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

(0) Obligation:

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

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

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

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 56 SCCs with 198 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

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.

ISPLNAT(ok(X)) → ISPLNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

ok₁ > ISPLNAT₁

The following usable rules [FROCOS05] were oriented: none

(7) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

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

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATURAL(ok(X)) → ISNATURAL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

ok₁ > ISNATURAL₁

The following usable rules [FROCOS05] were oriented: none

(12) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) Obligation:

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

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

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

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISLNAT(ok(X)) → ISLNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

ok₁ > ISLNAT₁

The following usable rules [FROCOS05] were oriented: none

(17) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(18) PisEmptyProof (EQUIVALENT transformation)

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

(19) TRUE

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

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

(21) 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(ok(X1), ok(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)
ok(x1)  =  ok(x1)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(22) Obligation:

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

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

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

(23) 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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(24) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(25) PisEmptyProof (EQUIVALENT transformation)

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

(26) TRUE

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

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

(28) 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(ok(X1), ok(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)
ok(x1)  =  ok(x1)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(29) Obligation:

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

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

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

(30) 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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(31) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(32) PisEmptyProof (EQUIVALENT transformation)

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

(33) TRUE

(34) Obligation:

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

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

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

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(36) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(mark(X)) → TAIL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > TAIL₁

The following usable rules [FROCOS05] were oriented: none

(38) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(39) PisEmptyProof (EQUIVALENT transformation)

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

(40) TRUE

(41) Obligation:

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

U91¹(ok(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(43) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(44) 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: Recursive Path Order [RPO].
Precedence:

mark₁ > U91^1₁

The following usable rules [FROCOS05] were oriented: none

(45) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(46) PisEmptyProof (EQUIVALENT transformation)

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

(47) TRUE

(48) Obligation:

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

U81¹(ok(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(50) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(51) 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: Recursive Path Order [RPO].
Precedence:

mark₁ > U81^1₁

The following usable rules [FROCOS05] were oriented: none

(52) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(53) PisEmptyProof (EQUIVALENT transformation)

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

(54) TRUE

(55) Obligation:

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

U71¹(ok(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(57) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(mark(X)) → U71¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > U71^1₁

The following usable rules [FROCOS05] were oriented: none

(59) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(60) PisEmptyProof (EQUIVALENT transformation)

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

(61) TRUE

(62) Obligation:

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

U61¹(ok(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(64) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(mark(X)) → U61¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > U61^1₁

The following usable rules [FROCOS05] were oriented: none

(66) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(67) PisEmptyProof (EQUIVALENT transformation)

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

(68) TRUE

(69) Obligation:

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

U52¹(ok(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(70) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(71) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(72) 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: Recursive Path Order [RPO].
Precedence:

mark₁ > U52^1₁

The following usable rules [FROCOS05] were oriented: none

(73) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(74) PisEmptyProof (EQUIVALENT transformation)

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

(75) TRUE

(76) Obligation:

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

U51¹(ok(X1), ok(X2)) → U51¹(X1, X2)
U51¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(ok(X1), ok(X2)) → U51¹(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

ok₁ > U51^1₁
mark₁ > U51^1₁

The following usable rules [FROCOS05] were oriented: none

(78) Obligation:

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

U51¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(79) 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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(80) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(81) PisEmptyProof (EQUIVALENT transformation)

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

(82) TRUE

(83) Obligation:

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

U42¹(ok(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(85) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(86) 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: Recursive Path Order [RPO].
Precedence:

mark₁ > U42^1₁

The following usable rules [FROCOS05] were oriented: none

(87) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(88) PisEmptyProof (EQUIVALENT transformation)

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

(89) TRUE

(90) Obligation:

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

U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)
U41¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U41^1₁
mark₁ > U41^1₁

The following usable rules [FROCOS05] were oriented: none

(92) Obligation:

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

U41¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(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)

Recursive Path Order [RPO].
Precedence:

trivial

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

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(ok(X1), ok(X2)) → U32¹(X1, X2)
U32¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(ok(X1), ok(X2)) → U32¹(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

ok₁ > U32^1₁
mark₁ > U32^1₁

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)

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

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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(101) Obligation:

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

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

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

(102) PisEmptyProof (EQUIVALENT transformation)

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

(103) TRUE

(104) Obligation:

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

U31¹(ok(X1), ok(X2), ok(X3)) → U31¹(X1, X2, X3)
U31¹(mark(X1), 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(105) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(ok(X1), ok(X2), ok(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(x3)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U31^1₁
mark₁ > U31^1₁

The following usable rules [FROCOS05] were oriented: none

(106) Obligation:

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

U31¹(mark(X1), 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(mark(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)  =  U31¹(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

mark₁ > U31^1₁

The following usable rules [FROCOS05] were oriented: none

(108) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(109) PisEmptyProof (EQUIVALENT transformation)

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

(110) TRUE

(111) Obligation:

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

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

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

(112) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FST(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(113) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(114) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FST(mark(X)) → FST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > FST₁

The following usable rules [FROCOS05] were oriented: none

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

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

(116) PisEmptyProof (EQUIVALENT transformation)

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

(117) TRUE

(118) Obligation:

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

U222¹(ok(X1), ok(X2), ok(X3)) → U222¹(X1, X2, X3)
U222¹(mark(X1), 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(119) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U222¹(ok(X1), ok(X2), ok(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(x3)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U222^1₁
mark₁ > U222^1₁

The following usable rules [FROCOS05] were oriented: none

(120) Obligation:

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

U222¹(mark(X1), 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(121) 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)

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

Recursive Path Order [RPO].
Precedence:

mark₁ > U222^1₁

The following usable rules [FROCOS05] were oriented: none

(122) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(123) PisEmptyProof (EQUIVALENT transformation)

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

(124) TRUE

(125) Obligation:

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

U221¹(ok(X1), ok(X2), ok(X3)) → U221¹(X1, X2, X3)
U221¹(mark(X1), 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(126) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U221¹(ok(X1), ok(X2), ok(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(x3)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U221^1₁
mark₁ > U221^1₁

The following usable rules [FROCOS05] were oriented: none

(127) Obligation:

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

U221¹(mark(X1), 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(128) 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)

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

Recursive Path Order [RPO].
Precedence:

mark₁ > U221^1₁

The following usable rules [FROCOS05] were oriented: none

(129) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(130) PisEmptyProof (EQUIVALENT transformation)

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

(131) TRUE

(132) Obligation:

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

U212¹(ok(X1), ok(X2)) → U212¹(X1, X2)
U212¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(133) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U212¹(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U212^1₁
mark₁ > U212^1₁

The following usable rules [FROCOS05] were oriented: none

(134) Obligation:

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

U212¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(135) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(136) Obligation:

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

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

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

(137) PisEmptyProof (EQUIVALENT transformation)

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

(138) TRUE

(139) Obligation:

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

U211¹(ok(X1), ok(X2)) → U211¹(X1, X2)
U211¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(140) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U211¹(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U211^1₁
mark₁ > U211^1₁

The following usable rules [FROCOS05] were oriented: none

(141) Obligation:

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

U211¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(142) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(143) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(144) PisEmptyProof (EQUIVALENT transformation)

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

(145) TRUE

(146) Obligation:

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

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

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

(147) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(ok(X1), ok(X2)) → U22¹(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

ok₁ > U22^1₁
mark₁ > U22^1₁

The following usable rules [FROCOS05] were oriented: none

(148) Obligation:

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

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

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

(149) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(150) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(151) PisEmptyProof (EQUIVALENT transformation)

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

(152) TRUE

(153) Obligation:

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

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

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.

U21¹(ok(X1), ok(X2), ok(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(x3)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U21^1₁
mark₁ > U21^1₁

The following usable rules [FROCOS05] were oriented: none

(155) Obligation:

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

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

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

(156) 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)

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

Recursive Path Order [RPO].
Precedence:

mark₁ > U21^1₁

The following usable rules [FROCOS05] were oriented: none

(157) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(158) PisEmptyProof (EQUIVALENT transformation)

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

(159) TRUE

(160) Obligation:

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

U204¹(ok(X1), ok(X2)) → U204¹(X1, X2)
U204¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U204¹(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U204^1₁
mark₁ > U204^1₁

The following usable rules [FROCOS05] were oriented: none

(162) Obligation:

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

U204¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

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)

Recursive Path Order [RPO].
Precedence:

trivial

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

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:

U203¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U203¹(X1, X2, X3, X4)
U203¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U203¹(ok(X1), ok(X2), ok(X3), ok(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(x4)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U203^1₁
mark₁ > U203^1₁

The following usable rules [FROCOS05] were oriented: none

(169) Obligation:

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

U203¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

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)  =  U203¹(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

mark₁ > U203^1₁

The following usable rules [FROCOS05] were oriented: none

(171) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(172) PisEmptyProof (EQUIVALENT transformation)

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

(173) TRUE

(174) Obligation:

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

U202¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U202¹(X1, X2, X3, X4)
U202¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(175) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U202¹(ok(X1), ok(X2), ok(X3), ok(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(x4)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U202^1₁
mark₁ > U202^1₁

The following usable rules [FROCOS05] were oriented: none

(176) Obligation:

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

U202¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

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)  =  U202¹(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

mark₁ > U202^1₁

The following usable rules [FROCOS05] were oriented: none

(178) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(179) PisEmptyProof (EQUIVALENT transformation)

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

(180) TRUE

(181) Obligation:

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

U201¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U201¹(X1, X2, X3, X4)
U201¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(182) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U201¹(ok(X1), ok(X2), ok(X3), ok(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(x4)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U201^1₁
mark₁ > U201^1₁

The following usable rules [FROCOS05] were oriented: none

(183) Obligation:

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

U201¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(184) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)  =  U201¹(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

mark₁ > U201^1₁

The following usable rules [FROCOS05] were oriented: none

(185) Obligation:

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

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

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

(186) PisEmptyProof (EQUIVALENT transformation)

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

(187) TRUE

(188) Obligation:

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

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

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

(189) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(X1, mark(X2)) → PAIR(X1, X2)
PAIR(ok(X1), ok(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)
ok(x1)  =  ok(x1)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(190) Obligation:

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

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

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

(191) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(192) Obligation:

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

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

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

(193) PisEmptyProof (EQUIVALENT transformation)

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

(194) TRUE

(195) Obligation:

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

U191¹(ok(X1), ok(X2)) → U191¹(X1, X2)
U191¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U191¹(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U191^1₁
mark₁ > U191^1₁

The following usable rules [FROCOS05] were oriented: none

(197) Obligation:

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

U191¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(198) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(199) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(200) PisEmptyProof (EQUIVALENT transformation)

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

(201) TRUE

(202) Obligation:

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

U182¹(ok(X1), ok(X2)) → U182¹(X1, X2)
U182¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U182¹(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U182^1₁
mark₁ > U182^1₁

The following usable rules [FROCOS05] were oriented: none

(204) Obligation:

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

U182¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(205) 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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(206) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(207) PisEmptyProof (EQUIVALENT transformation)

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

(208) TRUE

(209) Obligation:

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

U181¹(ok(X1), ok(X2)) → U181¹(X1, X2)
U181¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U181¹(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U181^1₁
mark₁ > U181^1₁

The following usable rules [FROCOS05] were oriented: none

(211) Obligation:

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

U181¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(213) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(214) PisEmptyProof (EQUIVALENT transformation)

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

(215) TRUE

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

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

(217) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
AFTERNTH(ok(X1), ok(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)
ok(x1)  =  ok(x1)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(218) Obligation:

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

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

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

(219) 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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(220) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(221) PisEmptyProof (EQUIVALENT transformation)

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

(222) TRUE

(223) Obligation:

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

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

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

(224) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(225) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(226) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(mark(X)) → HEAD(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > HEAD₁

The following usable rules [FROCOS05] were oriented: none

(227) Obligation:

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

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

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

(228) PisEmptyProof (EQUIVALENT transformation)

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

(229) TRUE

(230) Obligation:

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

U172¹(ok(X1), ok(X2), ok(X3)) → U172¹(X1, X2, X3)
U172¹(mark(X1), 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U172¹(ok(X1), ok(X2), ok(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(x3)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U172^1₁
mark₁ > U172^1₁

The following usable rules [FROCOS05] were oriented: none

(232) Obligation:

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

U172¹(mark(X1), 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U172¹(mark(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)  =  U172¹(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

mark₁ > U172^1₁

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

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:

U171¹(ok(X1), ok(X2), ok(X3)) → U171¹(X1, X2, X3)
U171¹(mark(X1), 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U171¹(ok(X1), ok(X2), ok(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(x3)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U171^1₁
mark₁ > U171^1₁

The following usable rules [FROCOS05] were oriented: none

(239) Obligation:

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

U171¹(mark(X1), 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U171¹(mark(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)  =  U171¹(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

mark₁ > U171^1₁

The following usable rules [FROCOS05] were oriented: none

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

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

(242) PisEmptyProof (EQUIVALENT transformation)

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

(243) TRUE

(244) Obligation:

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

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

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

(245) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(246) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > S₁

The following usable rules [FROCOS05] were oriented: none

(248) Obligation:

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

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

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

(249) PisEmptyProof (EQUIVALENT transformation)

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

(250) TRUE

(251) Obligation:

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

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

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

(252) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NATSFROM(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(253) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(254) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NATSFROM(mark(X)) → NATSFROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > NATSFROM₁

The following usable rules [FROCOS05] were oriented: none

(255) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(256) PisEmptyProof (EQUIVALENT transformation)

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

(257) TRUE

(258) Obligation:

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

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

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

(259) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(ok(X1), ok(X2)) → CONS(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

ok₁ > CONS₁
mark₁ > CONS₁

The following usable rules [FROCOS05] were oriented: none

(260) Obligation:

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

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

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

(261) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(262) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(263) PisEmptyProof (EQUIVALENT transformation)

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

(264) TRUE

(265) Obligation:

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

U161¹(ok(X1), ok(X2)) → U161¹(X1, X2)
U161¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(266) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U161¹(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U161^1₁
mark₁ > U161^1₁

The following usable rules [FROCOS05] were oriented: none

(267) Obligation:

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

U161¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(268) 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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(269) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(270) PisEmptyProof (EQUIVALENT transformation)

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

(271) TRUE

(272) Obligation:

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

U152¹(ok(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(273) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U152¹(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(274) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(275) 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: Recursive Path Order [RPO].
Precedence:

mark₁ > U152^1₁

The following usable rules [FROCOS05] were oriented: none

(276) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(277) PisEmptyProof (EQUIVALENT transformation)

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

(278) TRUE

(279) Obligation:

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

U151¹(ok(X1), ok(X2)) → U151¹(X1, X2)
U151¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(280) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U151¹(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U151^1₁
mark₁ > U151^1₁

The following usable rules [FROCOS05] were oriented: none

(281) Obligation:

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

U151¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(282) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(283) Obligation:

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

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

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

(284) PisEmptyProof (EQUIVALENT transformation)

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

(285) TRUE

(286) Obligation:

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

U142¹(ok(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(287) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U142¹(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(288) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(289) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U142¹(mark(X)) → U142¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > U142^1₁

The following usable rules [FROCOS05] were oriented: none

(290) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(291) PisEmptyProof (EQUIVALENT transformation)

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

(292) TRUE

(293) Obligation:

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

U141¹(ok(X1), ok(X2)) → U141¹(X1, X2)
U141¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U141¹(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U141^1₁
mark₁ > U141^1₁

The following usable rules [FROCOS05] were oriented: none

(295) Obligation:

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

U141¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(296) 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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(297) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(298) PisEmptyProof (EQUIVALENT transformation)

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

(299) TRUE

(300) Obligation:

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

U132¹(ok(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U132¹(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(302) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U132¹(mark(X)) → U132¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > U132^1₁

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

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:

U131¹(ok(X1), ok(X2)) → U131¹(X1, X2)
U131¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U131¹(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U131^1₁
mark₁ > U131^1₁

The following usable rules [FROCOS05] were oriented: none

(309) Obligation:

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

U131¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

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)

Recursive Path Order [RPO].
Precedence:

trivial

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

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:

U121¹(ok(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U121¹(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

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

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.

U121¹(mark(X)) → U121¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > U121^1₁

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

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:

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

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.

SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)
SPLITAT(ok(X1), ok(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)
ok(x1)  =  ok(x1)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(323) Obligation:

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

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

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.

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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

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

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

(326) PisEmptyProof (EQUIVALENT transformation)

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

(327) TRUE

(328) Obligation:

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

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

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

(329) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SND(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(330) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

SND(mark(X)) → SND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > SND₁

The following usable rules [FROCOS05] were oriented: none

(332) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(333) PisEmptyProof (EQUIVALENT transformation)

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

(334) TRUE

(335) Obligation:

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

U111¹(ok(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(336) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U111¹(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(337) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(338) 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: Recursive Path Order [RPO].
Precedence:

mark₁ > U111^1₁

The following usable rules [FROCOS05] were oriented: none

(339) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(340) PisEmptyProof (EQUIVALENT transformation)

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

(341) TRUE

(342) Obligation:

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

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

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

(343) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Recursive Path Order [RPO].
Precedence:

ok₁ > U12^1₁
mark₁ > U12^1₁

The following usable rules [FROCOS05] were oriented: none

(344) Obligation:

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

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

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

(345) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Recursive Path Order [RPO].
Precedence:

mark₁ > U12^1₁

The following usable rules [FROCOS05] were oriented: none

(346) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(347) PisEmptyProof (EQUIVALENT transformation)

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

(348) TRUE

(349) Obligation:

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

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

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

(350) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Recursive Path Order [RPO].
Precedence:

ok₁ > U11^1₁
mark₁ > U11^1₁

The following usable rules [FROCOS05] were oriented: none

(351) Obligation:

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

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

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

(352) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Recursive Path Order [RPO].
Precedence:

mark₁ > U11^1₁

The following usable rules [FROCOS05] were oriented: none

(353) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(354) PisEmptyProof (EQUIVALENT transformation)

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

(355) TRUE

(356) Obligation:

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

U102¹(ok(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(357) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(358) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(359) 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: Recursive Path Order [RPO].
Precedence:

mark₁ > U102^1₁

The following usable rules [FROCOS05] were oriented: none

(360) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(361) PisEmptyProof (EQUIVALENT transformation)

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

(362) TRUE

(363) Obligation:

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

U101¹(ok(X1), ok(X2)) → U101¹(X1, X2)
U101¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(364) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:

ok₁ > U101^1₁
mark₁ > U101^1₁

The following usable rules [FROCOS05] were oriented: none

(365) Obligation:

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

U101¹(mark(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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(366) 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)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(367) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(368) PisEmptyProof (EQUIVALENT transformation)

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

(369) TRUE

(370) Obligation:

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

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

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

(371) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

U101₂ > PROPER₁
U102₁ > PROPER₁
isLNat₁ > PROPER₁
U11₃ > PROPER₁
U12₃ > PROPER₁
U111₁ > PROPER₁
snd₁ > PROPER₁
splitAt₂ > PROPER₁
U121₁ > PROPER₁
U131₂ > PROPER₁
U132₁ > PROPER₁
U141₂ > PROPER₁
U142₁ > PROPER₁
U151₂ > PROPER₁
U152₁ > PROPER₁
U161₂ > PROPER₁
cons₂ > PROPER₁
natsFrom₁ > PROPER₁
s₁ > PROPER₁
U171₃ > PROPER₁
U172₃ > PROPER₁
head₁ > PROPER₁
afterNth₂ > PROPER₁
U181₂ > PROPER₁
U182₂ > PROPER₁
U191₂ > PROPER₁
pair₂ > PROPER₁
U201₄ > PROPER₁
U202₄ > PROPER₁
isNatural₁ > PROPER₁
U203₄ > PROPER₁
U204₂ > PROPER₁
U21₃ > PROPER₁
U22₂ > PROPER₁
U211₂ > PROPER₁
U212₂ > PROPER₁
U221₃ > PROPER₁
U222₃ > PROPER₁
fst₁ > PROPER₁
U31₃ > PROPER₁
U32₂ > PROPER₁
U41₂ > PROPER₁
U42₁ > PROPER₁
U51₂ > PROPER₁
U52₁ > PROPER₁
U61₁ > PROPER₁
U71₁ > PROPER₁
U81₁ > PROPER₁
U91₁ > PROPER₁
isPLNat₁ > PROPER₁
tail₁ > PROPER₁
take₂ > PROPER₁
sel₂ > PROPER₁

The following usable rules [FROCOS05] were oriented: none

(372) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(373) PisEmptyProof (EQUIVALENT transformation)

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

(374) TRUE

(375) Obligation:

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

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

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

(376) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U101(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(snd(X)) → ACTIVE(X)
ACTIVE(splitAt(X1, X2)) → ACTIVE(X1)
ACTIVE(splitAt(X1, X2)) → ACTIVE(X2)
ACTIVE(U132(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U172(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(afterNth(X1, X2)) → ACTIVE(X1)
ACTIVE(afterNth(X1, X2)) → ACTIVE(X2)
ACTIVE(pair(X1, X2)) → ACTIVE(X1)
ACTIVE(pair(X1, X2)) → ACTIVE(X2)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U102(x1)  =  x1
U101(x1, x2)  =  U101(x1)
U11(x1, x2, x3)  =  U11(x1, x2)
U12(x1, x2, x3)  =  x1
U111(x1)  =  x1
snd(x1)  =  snd(x1)
splitAt(x1, x2)  =  splitAt(x1, x2)
U121(x1)  =  x1
U131(x1, x2)  =  x1
U132(x1)  =  U132(x1)
U141(x1, x2)  =  x1
U142(x1)  =  x1
U151(x1, x2)  =  x1
U152(x1)  =  x1
U161(x1, x2)  =  x1
cons(x1, x2)  =  x1
natsFrom(x1)  =  x1
s(x1)  =  s(x1)
U171(x1, x2, x3)  =  x1
U172(x1, x2, x3)  =  U172(x1)
head(x1)  =  x1
afterNth(x1, x2)  =  afterNth(x1, x2)
U181(x1, x2)  =  x1
U182(x1, x2)  =  x1
U191(x1, x2)  =  x1
pair(x1, x2)  =  pair(x1, x2)
U201(x1, x2, x3, x4)  =  x1
U202(x1, x2, x3, x4)  =  x1
U203(x1, x2, x3, x4)  =  x1
U204(x1, x2)  =  x1
U21(x1, x2, x3)  =  x1
U22(x1, x2)  =  x1
U211(x1, x2)  =  x1
U212(x1, x2)  =  x1
U221(x1, x2, x3)  =  x1
U222(x1, x2, x3)  =  x1
fst(x1)  =  x1
U31(x1, x2, x3)  =  x1
U32(x1, x2)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1, x2)  =  x1
U52(x1)  =  x1
U61(x1)  =  x1
U71(x1)  =  x1
U81(x1)  =  x1
U91(x1)  =  x1
tail(x1)  =  x1
take(x1, x2)  =  take(x1, x2)
sel(x1, x2)  =  sel(x1, x2)

Recursive Path Order [RPO].
Precedence:

U101₁ > ACTIVE₁
U11₂ > ACTIVE₁
snd₁ > ACTIVE₁
splitAt₂ > ACTIVE₁
U132₁ > ACTIVE₁
s₁ > ACTIVE₁
U172₁ > ACTIVE₁
afterNth₂ > ACTIVE₁
pair₂ > ACTIVE₁
take₂ > ACTIVE₁
sel₂ > ACTIVE₁

The following usable rules [FROCOS05] were oriented: none

(377) Obligation:

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

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

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

(378) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U102(X)) → ACTIVE(X)
ACTIVE(U12(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U111(X)) → ACTIVE(X)
ACTIVE(U121(X)) → ACTIVE(X)
ACTIVE(U131(X1, X2)) → ACTIVE(X1)
ACTIVE(U141(X1, X2)) → ACTIVE(X1)
ACTIVE(U142(X)) → ACTIVE(X)
ACTIVE(U151(X1, X2)) → ACTIVE(X1)
ACTIVE(U152(X)) → ACTIVE(X)
ACTIVE(U161(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(natsFrom(X)) → ACTIVE(X)
ACTIVE(U171(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(head(X)) → ACTIVE(X)
ACTIVE(U181(X1, X2)) → ACTIVE(X1)
ACTIVE(U182(X1, X2)) → ACTIVE(X1)
ACTIVE(U191(X1, X2)) → ACTIVE(X1)
ACTIVE(U201(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U202(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U203(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U204(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U22(X1, X2)) → ACTIVE(X1)
ACTIVE(U211(X1, X2)) → ACTIVE(X1)
ACTIVE(U212(X1, X2)) → ACTIVE(X1)
ACTIVE(U221(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U222(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(fst(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U32(X1, X2)) → ACTIVE(X1)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2)) → ACTIVE(X1)
ACTIVE(U52(X)) → ACTIVE(X)
ACTIVE(U61(X)) → ACTIVE(X)
ACTIVE(U71(X)) → ACTIVE(X)
ACTIVE(U81(X)) → ACTIVE(X)
ACTIVE(U91(X)) → ACTIVE(X)
ACTIVE(tail(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U102(x1)  =  U102(x1)
U12(x1, x2, x3)  =  U12(x1)
U111(x1)  =  U111(x1)
U121(x1)  =  U121(x1)
U131(x1, x2)  =  U131(x1)
U141(x1, x2)  =  U141(x1)
U142(x1)  =  U142(x1)
U151(x1, x2)  =  U151(x1)
U152(x1)  =  U152(x1)
U161(x1, x2)  =  U161(x1)
cons(x1, x2)  =  cons(x1)
natsFrom(x1)  =  natsFrom(x1)
U171(x1, x2, x3)  =  U171(x1)
head(x1)  =  head(x1)
U181(x1, x2)  =  U181(x1)
U182(x1, x2)  =  U182(x1)
U191(x1, x2)  =  U191(x1)
U201(x1, x2, x3, x4)  =  U201(x1)
U202(x1, x2, x3, x4)  =  U202(x1)
U203(x1, x2, x3, x4)  =  U203(x1)
U204(x1, x2)  =  U204(x1)
U21(x1, x2, x3)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U211(x1, x2)  =  U211(x1)
U212(x1, x2)  =  U212(x1)
U221(x1, x2, x3)  =  U221(x1)
U222(x1, x2, x3)  =  U222(x1)
fst(x1)  =  fst(x1)
U31(x1, x2, x3)  =  U31(x1)
U32(x1, x2)  =  U32(x1)
U41(x1, x2)  =  U41(x1)
U42(x1)  =  U42(x1)
U51(x1, x2)  =  U51(x1)
U52(x1)  =  U52(x1)
U61(x1)  =  U61(x1)
U71(x1)  =  U71(x1)
U81(x1)  =  U81(x1)
U91(x1)  =  U91(x1)
tail(x1)  =  tail(x1)

Recursive Path Order [RPO].
Precedence:

U102₁ > ACTIVE₁
U12₁ > ACTIVE₁
U111₁ > ACTIVE₁
U121₁ > ACTIVE₁
U131₁ > ACTIVE₁
U141₁ > ACTIVE₁
U142₁ > ACTIVE₁
U151₁ > ACTIVE₁
U152₁ > ACTIVE₁
U161₁ > ACTIVE₁
cons₁ > ACTIVE₁
natsFrom₁ > ACTIVE₁
U171₁ > ACTIVE₁
head₁ > ACTIVE₁
U181₁ > ACTIVE₁
U182₁ > ACTIVE₁
U191₁ > ACTIVE₁
U201₁ > ACTIVE₁
U202₁ > ACTIVE₁
U203₁ > ACTIVE₁
U204₁ > ACTIVE₁
U21₁ > ACTIVE₁
U22₁ > ACTIVE₁
U211₁ > ACTIVE₁
U212₁ > ACTIVE₁
U221₁ > ACTIVE₁
U222₁ > ACTIVE₁
fst₁ > ACTIVE₁
U31₁ > ACTIVE₁
U32₁ > ACTIVE₁
U41₁ > ACTIVE₁
U42₁ > ACTIVE₁
U51₁ > ACTIVE₁
U52₁ > ACTIVE₁
U61₁ > ACTIVE₁
U71₁ > ACTIVE₁
U81₁ > ACTIVE₁
U91₁ > ACTIVE₁
tail₁ > ACTIVE₁

The following usable rules [FROCOS05] were oriented: none

(379) 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))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(380) PisEmptyProof (EQUIVALENT transformation)

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

(381) TRUE

(382) Obligation:

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

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

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