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, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

ACTIVE(U101(tt, V1, V2)) → MARK(U102(isNatural(V1), V2))
ACTIVE(U101(tt, V1, V2)) → U102¹(isNatural(V1), V2)
ACTIVE(U101(tt, V1, V2)) → ISNATURAL(V1)
ACTIVE(U102(tt, V2)) → MARK(U103(isLNat(V2)))
ACTIVE(U102(tt, V2)) → U103¹(isLNat(V2))
ACTIVE(U102(tt, V2)) → ISLNAT(V2)
ACTIVE(U103(tt)) → MARK(tt)
ACTIVE(U11(tt, N, XS)) → MARK(snd(splitAt(N, XS)))
ACTIVE(U11(tt, N, XS)) → SND(splitAt(N, XS))
ACTIVE(U11(tt, N, XS)) → SPLITAT(N, XS)
ACTIVE(U111(tt, V1)) → MARK(U112(isLNat(V1)))
ACTIVE(U111(tt, V1)) → U112¹(isLNat(V1))
ACTIVE(U111(tt, V1)) → ISLNAT(V1)
ACTIVE(U112(tt)) → MARK(tt)
ACTIVE(U121(tt, V1)) → MARK(U122(isNatural(V1)))
ACTIVE(U121(tt, V1)) → U122¹(isNatural(V1))
ACTIVE(U121(tt, V1)) → ISNATURAL(V1)
ACTIVE(U122(tt)) → MARK(tt)
ACTIVE(U131(tt, V1, V2)) → MARK(U132(isNatural(V1), V2))
ACTIVE(U131(tt, V1, V2)) → U132¹(isNatural(V1), V2)
ACTIVE(U131(tt, V1, V2)) → ISNATURAL(V1)
ACTIVE(U132(tt, V2)) → MARK(U133(isLNat(V2)))
ACTIVE(U132(tt, V2)) → U133¹(isLNat(V2))
ACTIVE(U132(tt, V2)) → ISLNAT(V2)
ACTIVE(U133(tt)) → MARK(tt)
ACTIVE(U141(tt, V1, V2)) → MARK(U142(isLNat(V1), V2))
ACTIVE(U141(tt, V1, V2)) → U142¹(isLNat(V1), V2)
ACTIVE(U141(tt, V1, V2)) → ISLNAT(V1)
ACTIVE(U142(tt, V2)) → MARK(U143(isLNat(V2)))
ACTIVE(U142(tt, V2)) → U143¹(isLNat(V2))
ACTIVE(U142(tt, V2)) → ISLNAT(V2)
ACTIVE(U143(tt)) → MARK(tt)
ACTIVE(U151(tt, V1, V2)) → MARK(U152(isNatural(V1), V2))
ACTIVE(U151(tt, V1, V2)) → U152¹(isNatural(V1), V2)
ACTIVE(U151(tt, V1, V2)) → ISNATURAL(V1)
ACTIVE(U152(tt, V2)) → MARK(U153(isLNat(V2)))
ACTIVE(U152(tt, V2)) → U153¹(isLNat(V2))
ACTIVE(U152(tt, V2)) → ISLNAT(V2)
ACTIVE(U153(tt)) → MARK(tt)
ACTIVE(U161(tt, N)) → MARK(cons(N, natsFrom(s(N))))
ACTIVE(U161(tt, N)) → CONS(N, natsFrom(s(N)))
ACTIVE(U161(tt, N)) → NATSFROM(s(N))
ACTIVE(U161(tt, N)) → S(N)
ACTIVE(U171(tt, N, XS)) → MARK(head(afterNth(N, XS)))
ACTIVE(U171(tt, N, XS)) → HEAD(afterNth(N, XS))
ACTIVE(U171(tt, N, XS)) → AFTERNTH(N, XS)
ACTIVE(U181(tt, Y)) → MARK(Y)
ACTIVE(U191(tt, XS)) → MARK(pair(nil, XS))
ACTIVE(U191(tt, XS)) → PAIR(nil, XS)
ACTIVE(U201(tt, N, X, XS)) → MARK(U202(splitAt(N, XS), X))
ACTIVE(U201(tt, N, X, XS)) → U202¹(splitAt(N, XS), X)
ACTIVE(U201(tt, N, X, XS)) → SPLITAT(N, XS)
ACTIVE(U202(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
ACTIVE(U202(pair(YS, ZS), X)) → PAIR(cons(X, YS), ZS)
ACTIVE(U202(pair(YS, ZS), X)) → CONS(X, YS)
ACTIVE(U21(tt, X)) → MARK(X)
ACTIVE(U211(tt, XS)) → MARK(XS)
ACTIVE(U221(tt, N, XS)) → MARK(fst(splitAt(N, XS)))
ACTIVE(U221(tt, N, XS)) → FST(splitAt(N, XS))
ACTIVE(U221(tt, N, XS)) → SPLITAT(N, XS)
ACTIVE(U31(tt, N)) → MARK(N)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isNatural(V1), V2))
ACTIVE(U41(tt, V1, V2)) → U42¹(isNatural(V1), V2)
ACTIVE(U41(tt, V1, V2)) → ISNATURAL(V1)
ACTIVE(U42(tt, V2)) → MARK(U43(isLNat(V2)))
ACTIVE(U42(tt, V2)) → U43¹(isLNat(V2))
ACTIVE(U42(tt, V2)) → ISLNAT(V2)
ACTIVE(U43(tt)) → MARK(tt)
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNatural(V1), V2))
ACTIVE(U51(tt, V1, V2)) → U52¹(isNatural(V1), V2)
ACTIVE(U51(tt, V1, V2)) → ISNATURAL(V1)
ACTIVE(U52(tt, V2)) → MARK(U53(isLNat(V2)))
ACTIVE(U52(tt, V2)) → U53¹(isLNat(V2))
ACTIVE(U52(tt, V2)) → ISLNAT(V2)
ACTIVE(U53(tt)) → MARK(tt)
ACTIVE(U61(tt, V1)) → MARK(U62(isPLNat(V1)))
ACTIVE(U61(tt, V1)) → U62¹(isPLNat(V1))
ACTIVE(U61(tt, V1)) → ISPLNAT(V1)
ACTIVE(U62(tt)) → MARK(tt)
ACTIVE(U71(tt, V1)) → MARK(U72(isNatural(V1)))
ACTIVE(U71(tt, V1)) → U72¹(isNatural(V1))
ACTIVE(U71(tt, V1)) → ISNATURAL(V1)
ACTIVE(U72(tt)) → MARK(tt)
ACTIVE(U81(tt, V1)) → MARK(U82(isPLNat(V1)))
ACTIVE(U81(tt, V1)) → U82¹(isPLNat(V1))
ACTIVE(U81(tt, V1)) → ISPLNAT(V1)
ACTIVE(U82(tt)) → MARK(tt)
ACTIVE(U91(tt, V1)) → MARK(U92(isLNat(V1)))
ACTIVE(U91(tt, V1)) → U92¹(isLNat(V1))
ACTIVE(U91(tt, V1)) → ISLNAT(V1)
ACTIVE(U92(tt)) → MARK(tt)
ACTIVE(afterNth(N, XS)) → MARK(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
ACTIVE(afterNth(N, XS)) → U11¹(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)
ACTIVE(afterNth(N, XS)) → AND(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
ACTIVE(afterNth(N, XS)) → AND(isNatural(N), isNaturalKind(N))
ACTIVE(afterNth(N, XS)) → ISNATURAL(N)
ACTIVE(afterNth(N, XS)) → ISNATURALKIND(N)
ACTIVE(afterNth(N, XS)) → AND(isLNat(XS), isLNatKind(XS))
ACTIVE(afterNth(N, XS)) → ISLNAT(XS)
ACTIVE(afterNth(N, XS)) → ISLNATKIND(XS)
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(fst(pair(X, Y))) → MARK(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
ACTIVE(fst(pair(X, Y))) → U21¹(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X)
ACTIVE(fst(pair(X, Y))) → AND(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y)))
ACTIVE(fst(pair(X, Y))) → AND(isLNat(X), isLNatKind(X))
ACTIVE(fst(pair(X, Y))) → ISLNAT(X)
ACTIVE(fst(pair(X, Y))) → ISLNATKIND(X)
ACTIVE(fst(pair(X, Y))) → AND(isLNat(Y), isLNatKind(Y))
ACTIVE(fst(pair(X, Y))) → ISLNAT(Y)
ACTIVE(fst(pair(X, Y))) → ISLNATKIND(Y)
ACTIVE(head(cons(N, XS))) → MARK(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
ACTIVE(head(cons(N, XS))) → U31¹(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N)
ACTIVE(head(cons(N, XS))) → AND(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
ACTIVE(head(cons(N, XS))) → AND(isNatural(N), isNaturalKind(N))
ACTIVE(head(cons(N, XS))) → ISNATURAL(N)
ACTIVE(head(cons(N, XS))) → ISNATURALKIND(N)
ACTIVE(head(cons(N, XS))) → AND(isLNat(XS), isLNatKind(XS))
ACTIVE(head(cons(N, XS))) → ISLNAT(XS)
ACTIVE(head(cons(N, XS))) → ISLNATKIND(XS)
ACTIVE(isLNat(nil)) → MARK(tt)
ACTIVE(isLNat(afterNth(V1, V2))) → MARK(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
ACTIVE(isLNat(afterNth(V1, V2))) → U41¹(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)
ACTIVE(isLNat(afterNth(V1, V2))) → AND(isNaturalKind(V1), isLNatKind(V2))
ACTIVE(isLNat(afterNth(V1, V2))) → ISNATURALKIND(V1)
ACTIVE(isLNat(afterNth(V1, V2))) → ISLNATKIND(V2)
ACTIVE(isLNat(cons(V1, V2))) → MARK(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
ACTIVE(isLNat(cons(V1, V2))) → U51¹(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)
ACTIVE(isLNat(cons(V1, V2))) → AND(isNaturalKind(V1), isLNatKind(V2))
ACTIVE(isLNat(cons(V1, V2))) → ISNATURALKIND(V1)
ACTIVE(isLNat(cons(V1, V2))) → ISLNATKIND(V2)
ACTIVE(isLNat(fst(V1))) → MARK(U61(isPLNatKind(V1), V1))
ACTIVE(isLNat(fst(V1))) → U61¹(isPLNatKind(V1), V1)
ACTIVE(isLNat(fst(V1))) → ISPLNATKIND(V1)
ACTIVE(isLNat(natsFrom(V1))) → MARK(U71(isNaturalKind(V1), V1))
ACTIVE(isLNat(natsFrom(V1))) → U71¹(isNaturalKind(V1), V1)
ACTIVE(isLNat(natsFrom(V1))) → ISNATURALKIND(V1)
ACTIVE(isLNat(snd(V1))) → MARK(U81(isPLNatKind(V1), V1))
ACTIVE(isLNat(snd(V1))) → U81¹(isPLNatKind(V1), V1)
ACTIVE(isLNat(snd(V1))) → ISPLNATKIND(V1)
ACTIVE(isLNat(tail(V1))) → MARK(U91(isLNatKind(V1), V1))
ACTIVE(isLNat(tail(V1))) → U91¹(isLNatKind(V1), V1)
ACTIVE(isLNat(tail(V1))) → ISLNATKIND(V1)
ACTIVE(isLNat(take(V1, V2))) → MARK(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
ACTIVE(isLNat(take(V1, V2))) → U101¹(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)
ACTIVE(isLNat(take(V1, V2))) → AND(isNaturalKind(V1), isLNatKind(V2))
ACTIVE(isLNat(take(V1, V2))) → ISNATURALKIND(V1)
ACTIVE(isLNat(take(V1, V2))) → ISLNATKIND(V2)
ACTIVE(isLNatKind(nil)) → MARK(tt)
ACTIVE(isLNatKind(afterNth(V1, V2))) → MARK(and(isNaturalKind(V1), isLNatKind(V2)))
ACTIVE(isLNatKind(afterNth(V1, V2))) → AND(isNaturalKind(V1), isLNatKind(V2))
ACTIVE(isLNatKind(afterNth(V1, V2))) → ISNATURALKIND(V1)
ACTIVE(isLNatKind(afterNth(V1, V2))) → ISLNATKIND(V2)
ACTIVE(isLNatKind(cons(V1, V2))) → MARK(and(isNaturalKind(V1), isLNatKind(V2)))
ACTIVE(isLNatKind(cons(V1, V2))) → AND(isNaturalKind(V1), isLNatKind(V2))
ACTIVE(isLNatKind(cons(V1, V2))) → ISNATURALKIND(V1)
ACTIVE(isLNatKind(cons(V1, V2))) → ISLNATKIND(V2)
ACTIVE(isLNatKind(fst(V1))) → MARK(isPLNatKind(V1))
ACTIVE(isLNatKind(fst(V1))) → ISPLNATKIND(V1)
ACTIVE(isLNatKind(natsFrom(V1))) → MARK(isNaturalKind(V1))
ACTIVE(isLNatKind(natsFrom(V1))) → ISNATURALKIND(V1)
ACTIVE(isLNatKind(snd(V1))) → MARK(isPLNatKind(V1))
ACTIVE(isLNatKind(snd(V1))) → ISPLNATKIND(V1)
ACTIVE(isLNatKind(tail(V1))) → MARK(isLNatKind(V1))
ACTIVE(isLNatKind(tail(V1))) → ISLNATKIND(V1)
ACTIVE(isLNatKind(take(V1, V2))) → MARK(and(isNaturalKind(V1), isLNatKind(V2)))
ACTIVE(isLNatKind(take(V1, V2))) → AND(isNaturalKind(V1), isLNatKind(V2))
ACTIVE(isLNatKind(take(V1, V2))) → ISNATURALKIND(V1)
ACTIVE(isLNatKind(take(V1, V2))) → ISLNATKIND(V2)
ACTIVE(isNatural(0)) → MARK(tt)
ACTIVE(isNatural(head(V1))) → MARK(U111(isLNatKind(V1), V1))
ACTIVE(isNatural(head(V1))) → U111¹(isLNatKind(V1), V1)
ACTIVE(isNatural(head(V1))) → ISLNATKIND(V1)
ACTIVE(isNatural(s(V1))) → MARK(U121(isNaturalKind(V1), V1))
ACTIVE(isNatural(s(V1))) → U121¹(isNaturalKind(V1), V1)
ACTIVE(isNatural(s(V1))) → ISNATURALKIND(V1)
ACTIVE(isNatural(sel(V1, V2))) → MARK(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
ACTIVE(isNatural(sel(V1, V2))) → U131¹(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)
ACTIVE(isNatural(sel(V1, V2))) → AND(isNaturalKind(V1), isLNatKind(V2))
ACTIVE(isNatural(sel(V1, V2))) → ISNATURALKIND(V1)
ACTIVE(isNatural(sel(V1, V2))) → ISLNATKIND(V2)
ACTIVE(isNaturalKind(0)) → MARK(tt)
ACTIVE(isNaturalKind(head(V1))) → MARK(isLNatKind(V1))
ACTIVE(isNaturalKind(head(V1))) → ISLNATKIND(V1)
ACTIVE(isNaturalKind(s(V1))) → MARK(isNaturalKind(V1))
ACTIVE(isNaturalKind(s(V1))) → ISNATURALKIND(V1)
ACTIVE(isNaturalKind(sel(V1, V2))) → MARK(and(isNaturalKind(V1), isLNatKind(V2)))
ACTIVE(isNaturalKind(sel(V1, V2))) → AND(isNaturalKind(V1), isLNatKind(V2))
ACTIVE(isNaturalKind(sel(V1, V2))) → ISNATURALKIND(V1)
ACTIVE(isNaturalKind(sel(V1, V2))) → ISLNATKIND(V2)
ACTIVE(isPLNat(pair(V1, V2))) → MARK(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
ACTIVE(isPLNat(pair(V1, V2))) → U141¹(and(isLNatKind(V1), isLNatKind(V2)), V1, V2)
ACTIVE(isPLNat(pair(V1, V2))) → AND(isLNatKind(V1), isLNatKind(V2))
ACTIVE(isPLNat(pair(V1, V2))) → ISLNATKIND(V1)
ACTIVE(isPLNat(pair(V1, V2))) → ISLNATKIND(V2)
ACTIVE(isPLNat(splitAt(V1, V2))) → MARK(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
ACTIVE(isPLNat(splitAt(V1, V2))) → U151¹(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)
ACTIVE(isPLNat(splitAt(V1, V2))) → AND(isNaturalKind(V1), isLNatKind(V2))
ACTIVE(isPLNat(splitAt(V1, V2))) → ISNATURALKIND(V1)
ACTIVE(isPLNat(splitAt(V1, V2))) → ISLNATKIND(V2)
ACTIVE(isPLNatKind(pair(V1, V2))) → MARK(and(isLNatKind(V1), isLNatKind(V2)))
ACTIVE(isPLNatKind(pair(V1, V2))) → AND(isLNatKind(V1), isLNatKind(V2))
ACTIVE(isPLNatKind(pair(V1, V2))) → ISLNATKIND(V1)
ACTIVE(isPLNatKind(pair(V1, V2))) → ISLNATKIND(V2)
ACTIVE(isPLNatKind(splitAt(V1, V2))) → MARK(and(isNaturalKind(V1), isLNatKind(V2)))
ACTIVE(isPLNatKind(splitAt(V1, V2))) → AND(isNaturalKind(V1), isLNatKind(V2))
ACTIVE(isPLNatKind(splitAt(V1, V2))) → ISNATURALKIND(V1)
ACTIVE(isPLNatKind(splitAt(V1, V2))) → ISLNATKIND(V2)
ACTIVE(natsFrom(N)) → MARK(U161(and(isNatural(N), isNaturalKind(N)), N))
ACTIVE(natsFrom(N)) → U161¹(and(isNatural(N), isNaturalKind(N)), N)
ACTIVE(natsFrom(N)) → AND(isNatural(N), isNaturalKind(N))
ACTIVE(natsFrom(N)) → ISNATURAL(N)
ACTIVE(natsFrom(N)) → ISNATURALKIND(N)
ACTIVE(sel(N, XS)) → MARK(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
ACTIVE(sel(N, XS)) → U171¹(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)
ACTIVE(sel(N, XS)) → AND(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
ACTIVE(sel(N, XS)) → AND(isNatural(N), isNaturalKind(N))
ACTIVE(sel(N, XS)) → ISNATURAL(N)
ACTIVE(sel(N, XS)) → ISNATURALKIND(N)
ACTIVE(sel(N, XS)) → AND(isLNat(XS), isLNatKind(XS))
ACTIVE(sel(N, XS)) → ISLNAT(XS)
ACTIVE(sel(N, XS)) → ISLNATKIND(XS)
ACTIVE(snd(pair(X, Y))) → MARK(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
ACTIVE(snd(pair(X, Y))) → U181¹(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y)
ACTIVE(snd(pair(X, Y))) → AND(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y)))
ACTIVE(snd(pair(X, Y))) → AND(isLNat(X), isLNatKind(X))
ACTIVE(snd(pair(X, Y))) → ISLNAT(X)
ACTIVE(snd(pair(X, Y))) → ISLNATKIND(X)
ACTIVE(snd(pair(X, Y))) → AND(isLNat(Y), isLNatKind(Y))
ACTIVE(snd(pair(X, Y))) → ISLNAT(Y)
ACTIVE(snd(pair(X, Y))) → ISLNATKIND(Y)
ACTIVE(splitAt(0, XS)) → MARK(U191(and(isLNat(XS), isLNatKind(XS)), XS))
ACTIVE(splitAt(0, XS)) → U191¹(and(isLNat(XS), isLNatKind(XS)), XS)
ACTIVE(splitAt(0, XS)) → AND(isLNat(XS), isLNatKind(XS))
ACTIVE(splitAt(0, XS)) → ISLNAT(XS)
ACTIVE(splitAt(0, XS)) → ISLNATKIND(XS)
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
ACTIVE(splitAt(s(N), cons(X, XS))) → U201¹(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS)
ACTIVE(splitAt(s(N), cons(X, XS))) → AND(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS))))
ACTIVE(splitAt(s(N), cons(X, XS))) → AND(isNatural(N), isNaturalKind(N))
ACTIVE(splitAt(s(N), cons(X, XS))) → ISNATURAL(N)
ACTIVE(splitAt(s(N), cons(X, XS))) → ISNATURALKIND(N)
ACTIVE(splitAt(s(N), cons(X, XS))) → AND(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))
ACTIVE(splitAt(s(N), cons(X, XS))) → AND(isNatural(X), isNaturalKind(X))
ACTIVE(splitAt(s(N), cons(X, XS))) → ISNATURAL(X)
ACTIVE(splitAt(s(N), cons(X, XS))) → ISNATURALKIND(X)
ACTIVE(splitAt(s(N), cons(X, XS))) → AND(isLNat(XS), isLNatKind(XS))
ACTIVE(splitAt(s(N), cons(X, XS))) → ISLNAT(XS)
ACTIVE(splitAt(s(N), cons(X, XS))) → ISLNATKIND(XS)
ACTIVE(tail(cons(N, XS))) → MARK(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
ACTIVE(tail(cons(N, XS))) → U211¹(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS)
ACTIVE(tail(cons(N, XS))) → AND(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
ACTIVE(tail(cons(N, XS))) → AND(isNatural(N), isNaturalKind(N))
ACTIVE(tail(cons(N, XS))) → ISNATURAL(N)
ACTIVE(tail(cons(N, XS))) → ISNATURALKIND(N)
ACTIVE(tail(cons(N, XS))) → AND(isLNat(XS), isLNatKind(XS))
ACTIVE(tail(cons(N, XS))) → ISLNAT(XS)
ACTIVE(tail(cons(N, XS))) → ISLNATKIND(XS)
ACTIVE(take(N, XS)) → MARK(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
ACTIVE(take(N, XS)) → U221¹(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)
ACTIVE(take(N, XS)) → AND(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
ACTIVE(take(N, XS)) → AND(isNatural(N), isNaturalKind(N))
ACTIVE(take(N, XS)) → ISNATURAL(N)
ACTIVE(take(N, XS)) → ISNATURALKIND(N)
ACTIVE(take(N, XS)) → AND(isLNat(XS), isLNatKind(XS))
ACTIVE(take(N, XS)) → ISLNAT(XS)
ACTIVE(take(N, XS)) → ISLNATKIND(XS)
MARK(U101(X1, X2, X3)) → ACTIVE(U101(mark(X1), X2, X3))
MARK(U101(X1, X2, X3)) → U101¹(mark(X1), X2, X3)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(tt) → ACTIVE(tt)
MARK(U102(X1, X2)) → ACTIVE(U102(mark(X1), X2))
MARK(U102(X1, X2)) → U102¹(mark(X1), X2)
MARK(U102(X1, X2)) → MARK(X1)
MARK(isNatural(X)) → ACTIVE(isNatural(X))
MARK(U103(X)) → ACTIVE(U103(mark(X)))
MARK(U103(X)) → U103¹(mark(X))
MARK(U103(X)) → MARK(X)
MARK(isLNat(X)) → ACTIVE(isLNat(X))
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(U11(X1, X2, X3)) → U11¹(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(snd(X)) → SND(mark(X))
MARK(snd(X)) → MARK(X)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
MARK(splitAt(X1, X2)) → SPLITAT(mark(X1), mark(X2))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U111(X1, X2)) → ACTIVE(U111(mark(X1), X2))
MARK(U111(X1, X2)) → U111¹(mark(X1), X2)
MARK(U111(X1, X2)) → MARK(X1)
MARK(U112(X)) → ACTIVE(U112(mark(X)))
MARK(U112(X)) → U112¹(mark(X))
MARK(U112(X)) → MARK(X)
MARK(U121(X1, X2)) → ACTIVE(U121(mark(X1), X2))
MARK(U121(X1, X2)) → U121¹(mark(X1), X2)
MARK(U121(X1, X2)) → MARK(X1)
MARK(U122(X)) → ACTIVE(U122(mark(X)))
MARK(U122(X)) → U122¹(mark(X))
MARK(U122(X)) → MARK(X)
MARK(U131(X1, X2, X3)) → ACTIVE(U131(mark(X1), X2, X3))
MARK(U131(X1, X2, X3)) → U131¹(mark(X1), X2, X3)
MARK(U131(X1, X2, X3)) → MARK(X1)
MARK(U132(X1, X2)) → ACTIVE(U132(mark(X1), X2))
MARK(U132(X1, X2)) → U132¹(mark(X1), X2)
MARK(U132(X1, X2)) → MARK(X1)
MARK(U133(X)) → ACTIVE(U133(mark(X)))
MARK(U133(X)) → U133¹(mark(X))
MARK(U133(X)) → MARK(X)
MARK(U141(X1, X2, X3)) → ACTIVE(U141(mark(X1), X2, X3))
MARK(U141(X1, X2, X3)) → U141¹(mark(X1), X2, X3)
MARK(U141(X1, X2, X3)) → MARK(X1)
MARK(U142(X1, X2)) → ACTIVE(U142(mark(X1), X2))
MARK(U142(X1, X2)) → U142¹(mark(X1), X2)
MARK(U142(X1, X2)) → MARK(X1)
MARK(U143(X)) → ACTIVE(U143(mark(X)))
MARK(U143(X)) → U143¹(mark(X))
MARK(U143(X)) → MARK(X)
MARK(U151(X1, X2, X3)) → ACTIVE(U151(mark(X1), X2, X3))
MARK(U151(X1, X2, X3)) → U151¹(mark(X1), X2, X3)
MARK(U151(X1, X2, X3)) → MARK(X1)
MARK(U152(X1, X2)) → ACTIVE(U152(mark(X1), X2))
MARK(U152(X1, X2)) → U152¹(mark(X1), X2)
MARK(U152(X1, X2)) → MARK(X1)
MARK(U153(X)) → ACTIVE(U153(mark(X)))
MARK(U153(X)) → U153¹(mark(X))
MARK(U153(X)) → MARK(X)
MARK(U161(X1, X2)) → ACTIVE(U161(mark(X1), X2))
MARK(U161(X1, X2)) → U161¹(mark(X1), X2)
MARK(U161(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
MARK(natsFrom(X)) → NATSFROM(mark(X))
MARK(natsFrom(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(U171(X1, X2, X3)) → ACTIVE(U171(mark(X1), X2, X3))
MARK(U171(X1, X2, X3)) → U171¹(mark(X1), X2, X3)
MARK(U171(X1, X2, X3)) → MARK(X1)
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(head(X)) → HEAD(mark(X))
MARK(head(X)) → MARK(X)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
MARK(afterNth(X1, X2)) → AFTERNTH(mark(X1), mark(X2))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(U181(X1, X2)) → ACTIVE(U181(mark(X1), X2))
MARK(U181(X1, X2)) → U181¹(mark(X1), X2)
MARK(U181(X1, X2)) → MARK(X1)
MARK(U191(X1, X2)) → ACTIVE(U191(mark(X1), X2))
MARK(U191(X1, X2)) → U191¹(mark(X1), X2)
MARK(U191(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → ACTIVE(pair(mark(X1), mark(X2)))
MARK(pair(X1, X2)) → PAIR(mark(X1), mark(X2))
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
MARK(nil) → ACTIVE(nil)
MARK(U201(X1, X2, X3, X4)) → ACTIVE(U201(mark(X1), X2, X3, X4))
MARK(U201(X1, X2, X3, X4)) → U201¹(mark(X1), X2, X3, X4)
MARK(U201(X1, X2, X3, X4)) → MARK(X1)
MARK(U202(X1, X2)) → ACTIVE(U202(mark(X1), X2))
MARK(U202(X1, X2)) → U202¹(mark(X1), X2)
MARK(U202(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → U21¹(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U211(X1, X2)) → ACTIVE(U211(mark(X1), X2))
MARK(U211(X1, X2)) → U211¹(mark(X1), X2)
MARK(U211(X1, X2)) → MARK(X1)
MARK(U221(X1, X2, X3)) → ACTIVE(U221(mark(X1), X2, X3))
MARK(U221(X1, X2, X3)) → U221¹(mark(X1), X2, X3)
MARK(U221(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(fst(X)) → FST(mark(X))
MARK(fst(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U31(X1, X2)) → U31¹(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → U41¹(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → U42¹(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U43(X)) → ACTIVE(U43(mark(X)))
MARK(U43(X)) → U43¹(mark(X))
MARK(U43(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → U51¹(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U52(X1, X2)) → U52¹(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U53(X)) → ACTIVE(U53(mark(X)))
MARK(U53(X)) → U53¹(mark(X))
MARK(U53(X)) → MARK(X)
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U61(X1, X2)) → U61¹(mark(X1), X2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → ACTIVE(U62(mark(X)))
MARK(U62(X)) → U62¹(mark(X))
MARK(U62(X)) → MARK(X)
MARK(isPLNat(X)) → ACTIVE(isPLNat(X))
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U71(X1, X2)) → U71¹(mark(X1), X2)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → ACTIVE(U72(mark(X)))
MARK(U72(X)) → U72¹(mark(X))
MARK(U72(X)) → MARK(X)
MARK(U81(X1, X2)) → ACTIVE(U81(mark(X1), X2))
MARK(U81(X1, X2)) → U81¹(mark(X1), X2)
MARK(U81(X1, X2)) → MARK(X1)
MARK(U82(X)) → ACTIVE(U82(mark(X)))
MARK(U82(X)) → U82¹(mark(X))
MARK(U82(X)) → MARK(X)
MARK(U91(X1, X2)) → ACTIVE(U91(mark(X1), X2))
MARK(U91(X1, X2)) → U91¹(mark(X1), X2)
MARK(U91(X1, X2)) → MARK(X1)
MARK(U92(X)) → ACTIVE(U92(mark(X)))
MARK(U92(X)) → U92¹(mark(X))
MARK(U92(X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(and(X1, X2)) → AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNaturalKind(X)) → ACTIVE(isNaturalKind(X))
MARK(isLNatKind(X)) → ACTIVE(isLNatKind(X))
MARK(isPLNatKind(X)) → ACTIVE(isPLNatKind(X))
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(tail(X)) → TAIL(mark(X))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(0) → ACTIVE(0)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → SEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, mark(X2), X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, mark(X3)) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, active(X2), X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)
U102¹(mark(X1), X2) → U102¹(X1, X2)
U102¹(X1, mark(X2)) → U102¹(X1, X2)
U102¹(active(X1), X2) → U102¹(X1, X2)
U102¹(X1, active(X2)) → U102¹(X1, X2)
ISNATURAL(mark(X)) → ISNATURAL(X)
ISNATURAL(active(X)) → ISNATURAL(X)
U103¹(mark(X)) → U103¹(X)
U103¹(active(X)) → U103¹(X)
ISLNAT(mark(X)) → ISLNAT(X)
ISLNAT(active(X)) → ISLNAT(X)
U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)
SND(mark(X)) → SND(X)
SND(active(X)) → SND(X)
SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)
SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)
U111¹(mark(X1), X2) → U111¹(X1, X2)
U111¹(X1, mark(X2)) → U111¹(X1, X2)
U111¹(active(X1), X2) → U111¹(X1, X2)
U111¹(X1, active(X2)) → U111¹(X1, X2)
U112¹(mark(X)) → U112¹(X)
U112¹(active(X)) → U112¹(X)
U121¹(mark(X1), X2) → U121¹(X1, X2)
U121¹(X1, mark(X2)) → U121¹(X1, X2)
U121¹(active(X1), X2) → U121¹(X1, X2)
U121¹(X1, active(X2)) → U121¹(X1, X2)
U122¹(mark(X)) → U122¹(X)
U122¹(active(X)) → U122¹(X)
U131¹(mark(X1), X2, X3) → U131¹(X1, X2, X3)
U131¹(X1, mark(X2), X3) → U131¹(X1, X2, X3)
U131¹(X1, X2, mark(X3)) → U131¹(X1, X2, X3)
U131¹(active(X1), X2, X3) → U131¹(X1, X2, X3)
U131¹(X1, active(X2), X3) → U131¹(X1, X2, X3)
U131¹(X1, X2, active(X3)) → U131¹(X1, X2, X3)
U132¹(mark(X1), X2) → U132¹(X1, X2)
U132¹(X1, mark(X2)) → U132¹(X1, X2)
U132¹(active(X1), X2) → U132¹(X1, X2)
U132¹(X1, active(X2)) → U132¹(X1, X2)
U133¹(mark(X)) → U133¹(X)
U133¹(active(X)) → U133¹(X)
U141¹(mark(X1), X2, X3) → U141¹(X1, X2, X3)
U141¹(X1, mark(X2), X3) → U141¹(X1, X2, X3)
U141¹(X1, X2, mark(X3)) → U141¹(X1, X2, X3)
U141¹(active(X1), X2, X3) → U141¹(X1, X2, X3)
U141¹(X1, active(X2), X3) → U141¹(X1, X2, X3)
U141¹(X1, X2, active(X3)) → U141¹(X1, X2, X3)
U142¹(mark(X1), X2) → U142¹(X1, X2)
U142¹(X1, mark(X2)) → U142¹(X1, X2)
U142¹(active(X1), X2) → U142¹(X1, X2)
U142¹(X1, active(X2)) → U142¹(X1, X2)
U143¹(mark(X)) → U143¹(X)
U143¹(active(X)) → U143¹(X)
U151¹(mark(X1), X2, X3) → U151¹(X1, X2, X3)
U151¹(X1, mark(X2), X3) → U151¹(X1, X2, X3)
U151¹(X1, X2, mark(X3)) → U151¹(X1, X2, X3)
U151¹(active(X1), X2, X3) → U151¹(X1, X2, X3)
U151¹(X1, active(X2), X3) → U151¹(X1, X2, X3)
U151¹(X1, X2, active(X3)) → U151¹(X1, X2, X3)
U152¹(mark(X1), X2) → U152¹(X1, X2)
U152¹(X1, mark(X2)) → U152¹(X1, X2)
U152¹(active(X1), X2) → U152¹(X1, X2)
U152¹(X1, active(X2)) → U152¹(X1, X2)
U153¹(mark(X)) → U153¹(X)
U153¹(active(X)) → U153¹(X)
U161¹(mark(X1), X2) → U161¹(X1, X2)
U161¹(X1, mark(X2)) → U161¹(X1, X2)
U161¹(active(X1), X2) → U161¹(X1, X2)
U161¹(X1, active(X2)) → U161¹(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
NATSFROM(mark(X)) → NATSFROM(X)
NATSFROM(active(X)) → NATSFROM(X)
S(mark(X)) → S(X)
S(active(X)) → S(X)
U171¹(mark(X1), X2, X3) → U171¹(X1, X2, X3)
U171¹(X1, mark(X2), X3) → U171¹(X1, X2, X3)
U171¹(X1, X2, mark(X3)) → U171¹(X1, X2, X3)
U171¹(active(X1), X2, X3) → U171¹(X1, X2, X3)
U171¹(X1, active(X2), X3) → U171¹(X1, X2, X3)
U171¹(X1, X2, active(X3)) → U171¹(X1, X2, X3)
HEAD(mark(X)) → HEAD(X)
HEAD(active(X)) → HEAD(X)
AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)
U181¹(mark(X1), X2) → U181¹(X1, X2)
U181¹(X1, mark(X2)) → U181¹(X1, X2)
U181¹(active(X1), X2) → U181¹(X1, X2)
U181¹(X1, active(X2)) → U181¹(X1, X2)
U191¹(mark(X1), X2) → U191¹(X1, X2)
U191¹(X1, mark(X2)) → U191¹(X1, X2)
U191¹(active(X1), X2) → U191¹(X1, X2)
U191¹(X1, active(X2)) → U191¹(X1, X2)
PAIR(mark(X1), X2) → PAIR(X1, X2)
PAIR(X1, mark(X2)) → PAIR(X1, X2)
PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)
U201¹(mark(X1), X2, X3, X4) → U201¹(X1, X2, X3, X4)
U201¹(X1, mark(X2), X3, X4) → U201¹(X1, X2, X3, X4)
U201¹(X1, X2, mark(X3), X4) → U201¹(X1, X2, X3, X4)
U201¹(X1, X2, X3, mark(X4)) → U201¹(X1, X2, X3, X4)
U201¹(active(X1), X2, X3, X4) → U201¹(X1, X2, X3, X4)
U201¹(X1, active(X2), X3, X4) → U201¹(X1, X2, X3, X4)
U201¹(X1, X2, active(X3), X4) → U201¹(X1, X2, X3, X4)
U201¹(X1, X2, X3, active(X4)) → U201¹(X1, X2, X3, X4)
U202¹(mark(X1), X2) → U202¹(X1, X2)
U202¹(X1, mark(X2)) → U202¹(X1, X2)
U202¹(active(X1), X2) → U202¹(X1, X2)
U202¹(X1, active(X2)) → U202¹(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)
U211¹(mark(X1), X2) → U211¹(X1, X2)
U211¹(X1, mark(X2)) → U211¹(X1, X2)
U211¹(active(X1), X2) → U211¹(X1, X2)
U211¹(X1, active(X2)) → U211¹(X1, X2)
U221¹(mark(X1), X2, X3) → U221¹(X1, X2, X3)
U221¹(X1, mark(X2), X3) → U221¹(X1, X2, X3)
U221¹(X1, X2, mark(X3)) → U221¹(X1, X2, X3)
U221¹(active(X1), X2, X3) → U221¹(X1, X2, X3)
U221¹(X1, active(X2), X3) → U221¹(X1, X2, X3)
U221¹(X1, X2, active(X3)) → U221¹(X1, X2, X3)
FST(mark(X)) → FST(X)
FST(active(X)) → FST(X)
U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(X1, mark(X2)) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)
U42¹(mark(X1), X2) → U42¹(X1, X2)
U42¹(X1, mark(X2)) → U42¹(X1, X2)
U42¹(active(X1), X2) → U42¹(X1, X2)
U42¹(X1, active(X2)) → U42¹(X1, X2)
U43¹(mark(X)) → U43¹(X)
U43¹(active(X)) → U43¹(X)
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(X1, mark(X2), X3) → U51¹(X1, X2, X3)
U51¹(X1, X2, mark(X3)) → U51¹(X1, X2, X3)
U51¹(active(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(X1, active(X2), X3) → U51¹(X1, X2, X3)
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)
U52¹(mark(X1), X2) → U52¹(X1, X2)
U52¹(X1, mark(X2)) → U52¹(X1, X2)
U52¹(active(X1), X2) → U52¹(X1, X2)
U52¹(X1, active(X2)) → U52¹(X1, X2)
U53¹(mark(X)) → U53¹(X)
U53¹(active(X)) → U53¹(X)
U61¹(mark(X1), X2) → U61¹(X1, X2)
U61¹(X1, mark(X2)) → U61¹(X1, X2)
U61¹(active(X1), X2) → U61¹(X1, X2)
U61¹(X1, active(X2)) → U61¹(X1, X2)
U62¹(mark(X)) → U62¹(X)
U62¹(active(X)) → U62¹(X)
ISPLNAT(mark(X)) → ISPLNAT(X)
ISPLNAT(active(X)) → ISPLNAT(X)
U71¹(mark(X1), X2) → U71¹(X1, X2)
U71¹(X1, mark(X2)) → U71¹(X1, X2)
U71¹(active(X1), X2) → U71¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)
U72¹(mark(X)) → U72¹(X)
U72¹(active(X)) → U72¹(X)
U81¹(mark(X1), X2) → U81¹(X1, X2)
U81¹(X1, mark(X2)) → U81¹(X1, X2)
U81¹(active(X1), X2) → U81¹(X1, X2)
U81¹(X1, active(X2)) → U81¹(X1, X2)
U82¹(mark(X)) → U82¹(X)
U82¹(active(X)) → U82¹(X)
U91¹(mark(X1), X2) → U91¹(X1, X2)
U91¹(X1, mark(X2)) → U91¹(X1, X2)
U91¹(active(X1), X2) → U91¹(X1, X2)
U91¹(X1, active(X2)) → U91¹(X1, X2)
U92¹(mark(X)) → U92¹(X)
U92¹(active(X)) → U92¹(X)
AND(mark(X1), X2) → AND(X1, X2)
AND(X1, mark(X2)) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)
ISNATURALKIND(mark(X)) → ISNATURALKIND(X)
ISNATURALKIND(active(X)) → ISNATURALKIND(X)
ISLNATKIND(mark(X)) → ISLNATKIND(X)
ISLNATKIND(active(X)) → ISLNATKIND(X)
ISPLNATKIND(mark(X)) → ISPLNATKIND(X)
ISPLNATKIND(active(X)) → ISPLNATKIND(X)
TAIL(mark(X)) → TAIL(X)
TAIL(active(X)) → TAIL(X)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 61 SCCs with 259 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(X1, mark(X2)) → SEL(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(mark(X1), X2) → SEL(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > SEL₂

Status:

mark₁: [1]
SEL₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(X1, active(X2)) → SEL(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

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

SEL(active(X1), X2) → SEL(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(12) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(active(X1), X2) → SEL(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(13) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(14) PisEmptyProof (EQUIVALENT transformation)

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

(15) TRUE

(16) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(X1, mark(X2)) → TAKE(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(mark(X1), X2) → TAKE(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > TAKE₂

Status:

TAKE₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(X1, active(X2)) → TAKE(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(22) Obligation:

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

TAKE(active(X1), X2) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(active(X1), X2) → TAKE(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

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, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(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:

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(mark(X)) → TAIL(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > TAIL₁

Status:

mark₁: [1]
TAIL₁: [1]

The following usable rules [FROCOS05] were oriented: none

(29) Obligation:

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

TAIL(active(X)) → TAIL(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(active(X)) → TAIL(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

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, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(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:

ISPLNATKIND(active(X)) → ISPLNATKIND(X)
ISPLNATKIND(mark(X)) → ISPLNATKIND(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISPLNATKIND(mark(X)) → ISPLNATKIND(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > ISPLNATKIND₁

Status:

ISPLNATKIND₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(36) Obligation:

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

ISPLNATKIND(active(X)) → ISPLNATKIND(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISPLNATKIND(active(X)) → ISPLNATKIND(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

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, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(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:

ISLNATKIND(active(X)) → ISLNATKIND(X)
ISLNATKIND(mark(X)) → ISLNATKIND(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISLNATKIND(mark(X)) → ISLNATKIND(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > ISLNATKIND₁

Status:

mark₁: [1]
ISLNATKIND₁: [1]

The following usable rules [FROCOS05] were oriented: none

(43) Obligation:

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

ISLNATKIND(active(X)) → ISLNATKIND(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISLNATKIND(active(X)) → ISLNATKIND(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

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, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(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:

ISNATURALKIND(active(X)) → ISNATURALKIND(X)
ISNATURALKIND(mark(X)) → ISNATURALKIND(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATURALKIND(mark(X)) → ISNATURALKIND(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > ISNATURALKIND₁

Status:

mark₁: [1]
ISNATURALKIND₁: [1]

The following usable rules [FROCOS05] were oriented: none

(50) Obligation:

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

ISNATURALKIND(active(X)) → ISNATURALKIND(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATURALKIND(active(X)) → ISNATURALKIND(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

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, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(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:

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(X1, mark(X2)) → AND(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(57) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(mark(X1), X2) → AND(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > AND₂

Status:

AND₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(59) Obligation:

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

AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(X1, active(X2)) → AND(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(61) Obligation:

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

AND(active(X1), X2) → AND(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(62) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(active(X1), X2) → AND(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(63) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(64) PisEmptyProof (EQUIVALENT transformation)

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

(65) TRUE

(66) Obligation:

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

U92¹(active(X)) → U92¹(X)
U92¹(mark(X)) → U92¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(mark(X)) → U92¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1)  =  U92¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U92^1₁

Status:

mark₁: [1]
U92^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(68) Obligation:

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

U92¹(active(X)) → U92¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(69) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(active(X)) → U92¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(70) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(71) PisEmptyProof (EQUIVALENT transformation)

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

(72) TRUE

(73) Obligation:

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

U91¹(X1, mark(X2)) → U91¹(X1, X2)
U91¹(mark(X1), X2) → U91¹(X1, X2)
U91¹(active(X1), X2) → U91¹(X1, X2)
U91¹(X1, active(X2)) → U91¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(X1, mark(X2)) → U91¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(75) Obligation:

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

U91¹(mark(X1), X2) → U91¹(X1, X2)
U91¹(active(X1), X2) → U91¹(X1, X2)
U91¹(X1, active(X2)) → U91¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(mark(X1), X2) → U91¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2)  =  U91¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U91^1₂

Status:

U91^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(77) Obligation:

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

U91¹(active(X1), X2) → U91¹(X1, X2)
U91¹(X1, active(X2)) → U91¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(78) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(X1, active(X2)) → U91¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(79) Obligation:

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

U91¹(active(X1), X2) → U91¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(80) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(active(X1), X2) → U91¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(81) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(82) PisEmptyProof (EQUIVALENT transformation)

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

(83) TRUE

(84) Obligation:

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

U82¹(active(X)) → U82¹(X)
U82¹(mark(X)) → U82¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(85) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(mark(X)) → U82¹(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U82^1₁

Status:

mark₁: [1]
U82^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(86) Obligation:

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

U82¹(active(X)) → U82¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(87) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(active(X)) → U82¹(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(88) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(89) PisEmptyProof (EQUIVALENT transformation)

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

(90) TRUE

(91) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(92) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(X1, mark(X2)) → U81¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(93) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(94) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(mark(X1), X2) → U81¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U81^1₂

Status:

U81^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(95) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(96) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(X1, active(X2)) → U81¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(97) Obligation:

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

U81¹(active(X1), X2) → U81¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(98) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(active(X1), X2) → U81¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(99) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(100) PisEmptyProof (EQUIVALENT transformation)

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

(101) TRUE

(102) Obligation:

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

U72¹(active(X)) → U72¹(X)
U72¹(mark(X)) → U72¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(103) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(mark(X)) → U72¹(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U72^1₁

Status:

mark₁: [1]
U72^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(104) Obligation:

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

U72¹(active(X)) → U72¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(105) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(active(X)) → U72¹(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(106) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(107) PisEmptyProof (EQUIVALENT transformation)

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

(108) TRUE

(109) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(110) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(X1, mark(X2)) → U71¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(111) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(112) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(mark(X1), X2) → U71¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U71^1₂

Status:

U71^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(113) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(114) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(X1, active(X2)) → U71¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(115) Obligation:

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

U71¹(active(X1), X2) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(116) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(active(X1), X2) → U71¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(117) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(118) PisEmptyProof (EQUIVALENT transformation)

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

(119) TRUE

(120) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(121) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISPLNAT(mark(X)) → ISPLNAT(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > ISPLNAT₁

Status:

ISPLNAT₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(122) Obligation:

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

ISPLNAT(active(X)) → ISPLNAT(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(123) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISPLNAT(active(X)) → ISPLNAT(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(124) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(125) PisEmptyProof (EQUIVALENT transformation)

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

(126) TRUE

(127) Obligation:

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

U62¹(active(X)) → U62¹(X)
U62¹(mark(X)) → U62¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(128) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(mark(X)) → U62¹(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U62^1₁

Status:

mark₁: [1]
U62^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(129) Obligation:

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

U62¹(active(X)) → U62¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(130) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(active(X)) → U62¹(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(131) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(132) PisEmptyProof (EQUIVALENT transformation)

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

(133) TRUE

(134) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(135) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, mark(X2)) → U61¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(136) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(137) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(mark(X1), X2) → U61¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U61^1₂

Status:

U61^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(138) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(139) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, active(X2)) → U61¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(140) Obligation:

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

U61¹(active(X1), X2) → U61¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(141) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(active(X1), X2) → U61¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(142) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(143) PisEmptyProof (EQUIVALENT transformation)

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

(144) TRUE

(145) Obligation:

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

U53¹(active(X)) → U53¹(X)
U53¹(mark(X)) → U53¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(146) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U53¹(mark(X)) → U53¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U53¹(x1)  =  U53¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U53^1₁

Status:

mark₁: [1]
U53^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(147) Obligation:

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

U53¹(active(X)) → U53¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(148) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U53¹(active(X)) → U53¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U53¹(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(149) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(150) PisEmptyProof (EQUIVALENT transformation)

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

(151) TRUE

(152) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(153) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(X1, mark(X2)) → U52¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(154) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(155) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(mark(X1), X2) → U52¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U52^1₂

Status:

U52^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(156) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(157) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(X1, active(X2)) → U52¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(158) Obligation:

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

U52¹(active(X1), X2) → U52¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(159) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(active(X1), X2) → U52¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(160) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(161) PisEmptyProof (EQUIVALENT transformation)

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

(162) TRUE

(163) Obligation:

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

U51¹(X1, mark(X2), X3) → U51¹(X1, X2, X3)
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(X1, X2, mark(X3)) → U51¹(X1, X2, X3)
U51¹(active(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(X1, active(X2), X3) → U51¹(X1, X2, X3)
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(164) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(X1, mark(X2), X3) → U51¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2, x3)  =  U51¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]
U51^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(165) Obligation:

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

U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(X1, X2, mark(X3)) → U51¹(X1, X2, X3)
U51¹(active(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(X1, active(X2), X3) → U51¹(X1, X2, X3)
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(166) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(X1, active(X2), X3) → U51¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2, x3)  =  U51¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U51^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(167) Obligation:

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

U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(X1, X2, mark(X3)) → U51¹(X1, X2, X3)
U51¹(active(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(168) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(X1, X2, mark(X3)) → U51¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(169) Obligation:

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

U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(active(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(170) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(active(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(X1, X2, active(X3)) → U51¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2, x3)  =  U51¹(x1, x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U51^1₂

Status:

active₁: [1]
U51^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(171) Obligation:

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

U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(172) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2, x3)  =  U51¹(x1, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U51^1₂

Status:

U51^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(173) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(174) PisEmptyProof (EQUIVALENT transformation)

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

(175) TRUE

(176) Obligation:

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

U43¹(active(X)) → U43¹(X)
U43¹(mark(X)) → U43¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(177) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U43¹(mark(X)) → U43¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U43¹(x1)  =  U43¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U43^1₁

Status:

mark₁: [1]
U43^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(178) Obligation:

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

U43¹(active(X)) → U43¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(179) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U43¹(active(X)) → U43¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U43¹(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(180) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(181) PisEmptyProof (EQUIVALENT transformation)

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

(182) TRUE

(183) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(184) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(X1, mark(X2)) → U42¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(185) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(186) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(mark(X1), X2) → U42¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U42^1₂

Status:

U42^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(187) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(188) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(X1, active(X2)) → U42¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(189) Obligation:

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

U42¹(active(X1), X2) → U42¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(190) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(active(X1), X2) → U42¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(191) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(192) PisEmptyProof (EQUIVALENT transformation)

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

(193) TRUE

(194) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(195) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U41^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(196) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(197) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U41^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(198) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(199) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(200) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(201) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

active₁ > U41^1₂

Status:

active₁: [1]
U41^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(202) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(203) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U41^1₂

Status:

U41^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(204) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(205) PisEmptyProof (EQUIVALENT transformation)

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

(206) TRUE

(207) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(208) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(X1, mark(X2)) → U31¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(209) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(210) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(mark(X1), X2) → U31¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U31^1₂

Status:

U31^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(211) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(212) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(X1, active(X2)) → U31¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(213) Obligation:

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

U31¹(active(X1), X2) → U31¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(214) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(active(X1), X2) → U31¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(215) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(216) PisEmptyProof (EQUIVALENT transformation)

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

(217) TRUE

(218) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(219) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FST(mark(X)) → FST(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > FST₁

Status:

FST₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(220) Obligation:

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

FST(active(X)) → FST(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(221) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FST(active(X)) → FST(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(222) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(223) PisEmptyProof (EQUIVALENT transformation)

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

(224) TRUE

(225) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(226) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]
U221^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(227) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(228) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U221^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(229) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(230) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(231) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(232) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

active₁ > U221^1₂

Status:

active₁: [1]
U221^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(233) 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, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(234) 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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U221^1₂

Status:

mark₁: [1]
U221^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(235) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(236) PisEmptyProof (EQUIVALENT transformation)

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

(237) TRUE

(238) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(239) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U211¹(X1, mark(X2)) → U211¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(240) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(241) 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)  =  U211¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U211^1₂

Status:

mark₁: [1]
U211^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(242) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(243) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U211¹(X1, active(X2)) → U211¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(244) Obligation:

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

U211¹(active(X1), X2) → U211¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(245) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U211¹(active(X1), X2) → U211¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(246) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(247) PisEmptyProof (EQUIVALENT transformation)

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

(248) TRUE

(249) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(250) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, mark(X2)) → U21¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(251) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(252) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(mark(X1), X2) → U21¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U21^1₂

Status:

U21^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(253) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(254) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, active(X2)) → U21¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(255) Obligation:

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

U21¹(active(X1), X2) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(256) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(active(X1), X2) → U21¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(257) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(258) PisEmptyProof (EQUIVALENT transformation)

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

(259) TRUE

(260) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(261) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U202¹(X1, mark(X2)) → U202¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(262) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(263) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U202¹(mark(X1), X2) → U202¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U202^1₂

Status:

U202^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(264) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(265) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U202¹(X1, active(X2)) → U202¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(266) Obligation:

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

U202¹(active(X1), X2) → U202¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(267) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U202¹(active(X1), X2) → U202¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(268) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(269) PisEmptyProof (EQUIVALENT transformation)

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

(270) TRUE

(271) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(272) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

active₁ > U201^1₁

Status:

U201^1₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(273) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(274) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U201¹(X1, X2, mark(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(x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U201^1₁

Status:

U201^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(275) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(276) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U201¹(X1, X2, X3, mark(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)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U201^1₁

Status:

U201^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(277) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(278) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

active₁ > U201^1₁

Status:

U201^1₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(279) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(280) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U201¹(X1, mark(X2), X3, X4) → U201¹(X1, X2, X3, X4)
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, x2, x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U201^1₃

Status:

mark₁: [1]
U201^1₃: [1,3,2]

The following usable rules [FROCOS05] were oriented: none

(281) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(282) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(283) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(284) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

active₁ > U201^1₂

Status:

active₁: [1]
U201^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(285) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(286) PisEmptyProof (EQUIVALENT transformation)

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

(287) TRUE

(288) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(289) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(X1, mark(X2)) → PAIR(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(290) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(291) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(mark(X1), X2) → PAIR(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > PAIR₂

Status:

PAIR₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(292) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(293) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(X1, active(X2)) → PAIR(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(294) Obligation:

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

PAIR(active(X1), X2) → PAIR(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(295) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(active(X1), X2) → PAIR(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(296) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(297) PisEmptyProof (EQUIVALENT transformation)

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

(298) TRUE

(299) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(300) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U191¹(X1, mark(X2)) → U191¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(301) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(302) 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)  =  U191¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U191^1₂

Status:

U191^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(303) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(304) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U191¹(X1, active(X2)) → U191¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(305) Obligation:

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

U191¹(active(X1), X2) → U191¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(306) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U191¹(active(X1), X2) → U191¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(307) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(308) PisEmptyProof (EQUIVALENT transformation)

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

(309) TRUE

(310) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(311) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U181¹(X1, mark(X2)) → U181¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(312) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(313) 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)  =  U181¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U181^1₂

Status:

U181^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(314) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(315) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U181¹(X1, active(X2)) → U181¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(316) Obligation:

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

U181¹(active(X1), X2) → U181¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(317) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U181¹(active(X1), X2) → U181¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

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, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(319) PisEmptyProof (EQUIVALENT transformation)

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

(320) TRUE

(321) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(322) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(323) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(324) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > AFTERNTH₂

Status:

mark₁: [1]
AFTERNTH₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(325) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(326) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(327) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(328) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(329) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(330) PisEmptyProof (EQUIVALENT transformation)

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

(331) TRUE

(332) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(333) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(mark(X)) → HEAD(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > HEAD₁

Status:

mark₁: [1]
HEAD₁: [1]

The following usable rules [FROCOS05] were oriented: none

(334) Obligation:

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

HEAD(active(X)) → HEAD(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(335) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(active(X)) → HEAD(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(336) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(337) PisEmptyProof (EQUIVALENT transformation)

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

(338) TRUE

(339) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(340) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]
U171^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(341) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(342) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U171^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(343) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(344) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(345) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(346) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

active₁ > U171^1₂

Status:

active₁: [1]
U171^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(347) 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, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(348) 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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U171^1₂

Status:

U171^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(349) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(350) PisEmptyProof (EQUIVALENT transformation)

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

(351) TRUE

(352) Obligation:

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(353) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > S₁

Status:

mark₁: [1]
S₁: [1]

The following usable rules [FROCOS05] were oriented: none

(354) Obligation:

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

S(active(X)) → S(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(355) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(356) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(357) PisEmptyProof (EQUIVALENT transformation)

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

(358) TRUE

(359) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(360) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NATSFROM(mark(X)) → NATSFROM(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > NATSFROM₁

Status:

mark₁: [1]
NATSFROM₁: [1]

The following usable rules [FROCOS05] were oriented: none

(361) Obligation:

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

NATSFROM(active(X)) → NATSFROM(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(362) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NATSFROM(active(X)) → NATSFROM(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(363) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(364) PisEmptyProof (EQUIVALENT transformation)

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

(365) TRUE

(366) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(367) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, mark(X2)) → CONS(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(368) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(369) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > CONS₂

Status:

CONS₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(370) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(371) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, active(X2)) → CONS(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(372) Obligation:

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

CONS(active(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(373) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(active(X1), X2) → CONS(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(374) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(375) PisEmptyProof (EQUIVALENT transformation)

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

(376) TRUE

(377) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(378) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U161¹(X1, mark(X2)) → U161¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(379) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(380) 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)  =  U161¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U161^1₂

Status:

mark₁: [1]
U161^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(381) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(382) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U161¹(X1, active(X2)) → U161¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(383) Obligation:

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

U161¹(active(X1), X2) → U161¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(384) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U161¹(active(X1), X2) → U161¹(X1, X2)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(385) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(386) PisEmptyProof (EQUIVALENT transformation)

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

(387) TRUE

(388) Obligation:

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

U153¹(active(X)) → U153¹(X)
U153¹(mark(X)) → U153¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(389) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U153¹(mark(X)) → U153¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U153¹(x1)  =  U153¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U153^1₁

Status:

mark₁: [1]
U153^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(390) Obligation:

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

U153¹(active(X)) → U153¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(391) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U153¹(active(X)) → U153¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U153¹(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(392) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(393) PisEmptyProof (EQUIVALENT transformation)

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

(394) TRUE

(395) Obligation:

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

U152¹(X1, mark(X2)) → U152¹(X1, X2)
U152¹(mark(X1), X2) → U152¹(X1, X2)
U152¹(active(X1), X2) → U152¹(X1, X2)
U152¹(X1, active(X2)) → U152¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(396) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U152¹(X1, mark(X2)) → U152¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U152¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(397) Obligation:

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

U152¹(mark(X1), X2) → U152¹(X1, X2)
U152¹(active(X1), X2) → U152¹(X1, X2)
U152¹(X1, active(X2)) → U152¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(398) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U152¹(mark(X1), X2) → U152¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U152¹(x1, x2)  =  U152¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U152^1₂

Status:

mark₁: [1]
U152^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(399) Obligation:

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

U152¹(active(X1), X2) → U152¹(X1, X2)
U152¹(X1, active(X2)) → U152¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(400) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U152¹(X1, active(X2)) → U152¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U152¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(401) Obligation:

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

U152¹(active(X1), X2) → U152¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(402) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U152¹(active(X1), X2) → U152¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U152¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(403) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(404) PisEmptyProof (EQUIVALENT transformation)

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

(405) TRUE

(406) Obligation:

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

U151¹(X1, mark(X2), X3) → U151¹(X1, X2, X3)
U151¹(mark(X1), X2, X3) → U151¹(X1, X2, X3)
U151¹(X1, X2, mark(X3)) → U151¹(X1, X2, X3)
U151¹(active(X1), X2, X3) → U151¹(X1, X2, X3)
U151¹(X1, active(X2), X3) → U151¹(X1, X2, X3)
U151¹(X1, X2, active(X3)) → U151¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(407) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U151¹(X1, mark(X2), X3) → U151¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U151¹(x1, x2, x3)  =  U151¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]
U151^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(408) Obligation:

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

U151¹(mark(X1), X2, X3) → U151¹(X1, X2, X3)
U151¹(X1, X2, mark(X3)) → U151¹(X1, X2, X3)
U151¹(active(X1), X2, X3) → U151¹(X1, X2, X3)
U151¹(X1, active(X2), X3) → U151¹(X1, X2, X3)
U151¹(X1, X2, active(X3)) → U151¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(409) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U151¹(X1, active(X2), X3) → U151¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U151¹(x1, x2, x3)  =  U151¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U151^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(410) Obligation:

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

U151¹(mark(X1), X2, X3) → U151¹(X1, X2, X3)
U151¹(X1, X2, mark(X3)) → U151¹(X1, X2, X3)
U151¹(active(X1), X2, X3) → U151¹(X1, X2, X3)
U151¹(X1, X2, active(X3)) → U151¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(411) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U151¹(X1, X2, mark(X3)) → U151¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U151¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(412) Obligation:

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

U151¹(mark(X1), X2, X3) → U151¹(X1, X2, X3)
U151¹(active(X1), X2, X3) → U151¹(X1, X2, X3)
U151¹(X1, X2, active(X3)) → U151¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(413) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U151¹(active(X1), X2, X3) → U151¹(X1, X2, X3)
U151¹(X1, X2, active(X3)) → U151¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U151¹(x1, x2, x3)  =  U151¹(x1, x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U151^1₂

Status:

active₁: [1]
U151^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(414) Obligation:

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

U151¹(mark(X1), X2, X3) → U151¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(415) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U151¹(mark(X1), X2, X3) → U151¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U151¹(x1, x2, x3)  =  U151¹(x1, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U151^1₂

Status:

mark₁: [1]
U151^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(416) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(417) PisEmptyProof (EQUIVALENT transformation)

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

(418) TRUE

(419) Obligation:

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

U143¹(active(X)) → U143¹(X)
U143¹(mark(X)) → U143¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(420) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U143¹(mark(X)) → U143¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U143¹(x1)  =  U143¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U143^1₁

Status:

mark₁: [1]
U143^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(421) Obligation:

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

U143¹(active(X)) → U143¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(422) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U143¹(active(X)) → U143¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U143¹(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(423) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(424) PisEmptyProof (EQUIVALENT transformation)

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

(425) TRUE

(426) Obligation:

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

U142¹(X1, mark(X2)) → U142¹(X1, X2)
U142¹(mark(X1), X2) → U142¹(X1, X2)
U142¹(active(X1), X2) → U142¹(X1, X2)
U142¹(X1, active(X2)) → U142¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(427) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U142¹(X1, mark(X2)) → U142¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U142¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(428) Obligation:

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

U142¹(mark(X1), X2) → U142¹(X1, X2)
U142¹(active(X1), X2) → U142¹(X1, X2)
U142¹(X1, active(X2)) → U142¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(429) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U142¹(mark(X1), X2) → U142¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U142¹(x1, x2)  =  U142¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U142^1₂

Status:

mark₁: [1]
U142^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(430) Obligation:

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

U142¹(active(X1), X2) → U142¹(X1, X2)
U142¹(X1, active(X2)) → U142¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(431) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U142¹(X1, active(X2)) → U142¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U142¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(432) Obligation:

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

U142¹(active(X1), X2) → U142¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(433) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U142¹(active(X1), X2) → U142¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U142¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(434) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(435) PisEmptyProof (EQUIVALENT transformation)

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

(436) TRUE

(437) Obligation:

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

U141¹(X1, mark(X2), X3) → U141¹(X1, X2, X3)
U141¹(mark(X1), X2, X3) → U141¹(X1, X2, X3)
U141¹(X1, X2, mark(X3)) → U141¹(X1, X2, X3)
U141¹(active(X1), X2, X3) → U141¹(X1, X2, X3)
U141¹(X1, active(X2), X3) → U141¹(X1, X2, X3)
U141¹(X1, X2, active(X3)) → U141¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(438) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U141¹(X1, mark(X2), X3) → U141¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U141¹(x1, x2, x3)  =  U141¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]
U141^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(439) Obligation:

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

U141¹(mark(X1), X2, X3) → U141¹(X1, X2, X3)
U141¹(X1, X2, mark(X3)) → U141¹(X1, X2, X3)
U141¹(active(X1), X2, X3) → U141¹(X1, X2, X3)
U141¹(X1, active(X2), X3) → U141¹(X1, X2, X3)
U141¹(X1, X2, active(X3)) → U141¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(440) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U141¹(X1, active(X2), X3) → U141¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U141¹(x1, x2, x3)  =  U141¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U141^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(441) Obligation:

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

U141¹(mark(X1), X2, X3) → U141¹(X1, X2, X3)
U141¹(X1, X2, mark(X3)) → U141¹(X1, X2, X3)
U141¹(active(X1), X2, X3) → U141¹(X1, X2, X3)
U141¹(X1, X2, active(X3)) → U141¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(442) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U141¹(X1, X2, mark(X3)) → U141¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U141¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(443) Obligation:

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

U141¹(mark(X1), X2, X3) → U141¹(X1, X2, X3)
U141¹(active(X1), X2, X3) → U141¹(X1, X2, X3)
U141¹(X1, X2, active(X3)) → U141¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(444) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U141¹(active(X1), X2, X3) → U141¹(X1, X2, X3)
U141¹(X1, X2, active(X3)) → U141¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U141¹(x1, x2, x3)  =  U141¹(x1, x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U141^1₂

Status:

active₁: [1]
U141^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(445) Obligation:

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

U141¹(mark(X1), X2, X3) → U141¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(446) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U141¹(mark(X1), X2, X3) → U141¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U141¹(x1, x2, x3)  =  U141¹(x1, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U141^1₂

Status:

U141^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(447) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(448) PisEmptyProof (EQUIVALENT transformation)

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

(449) TRUE

(450) Obligation:

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

U133¹(active(X)) → U133¹(X)
U133¹(mark(X)) → U133¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(451) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U133¹(mark(X)) → U133¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U133¹(x1)  =  U133¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U133^1₁

Status:

U133^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(452) Obligation:

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

U133¹(active(X)) → U133¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(453) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U133¹(active(X)) → U133¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U133¹(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(454) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(455) PisEmptyProof (EQUIVALENT transformation)

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

(456) TRUE

(457) Obligation:

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

U132¹(X1, mark(X2)) → U132¹(X1, X2)
U132¹(mark(X1), X2) → U132¹(X1, X2)
U132¹(active(X1), X2) → U132¹(X1, X2)
U132¹(X1, active(X2)) → U132¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(458) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U132¹(X1, mark(X2)) → U132¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U132¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(459) Obligation:

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

U132¹(mark(X1), X2) → U132¹(X1, X2)
U132¹(active(X1), X2) → U132¹(X1, X2)
U132¹(X1, active(X2)) → U132¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(460) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U132¹(mark(X1), X2) → U132¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U132¹(x1, x2)  =  U132¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U132^1₂

Status:

U132^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(461) Obligation:

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

U132¹(active(X1), X2) → U132¹(X1, X2)
U132¹(X1, active(X2)) → U132¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(462) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U132¹(X1, active(X2)) → U132¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U132¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(463) Obligation:

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

U132¹(active(X1), X2) → U132¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(464) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U132¹(active(X1), X2) → U132¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U132¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(465) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(466) PisEmptyProof (EQUIVALENT transformation)

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

(467) TRUE

(468) Obligation:

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

U131¹(X1, mark(X2), X3) → U131¹(X1, X2, X3)
U131¹(mark(X1), X2, X3) → U131¹(X1, X2, X3)
U131¹(X1, X2, mark(X3)) → U131¹(X1, X2, X3)
U131¹(active(X1), X2, X3) → U131¹(X1, X2, X3)
U131¹(X1, active(X2), X3) → U131¹(X1, X2, X3)
U131¹(X1, X2, active(X3)) → U131¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(469) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U131¹(X1, mark(X2), X3) → U131¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131¹(x1, x2, x3)  =  U131¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U131^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(470) Obligation:

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

U131¹(mark(X1), X2, X3) → U131¹(X1, X2, X3)
U131¹(X1, X2, mark(X3)) → U131¹(X1, X2, X3)
U131¹(active(X1), X2, X3) → U131¹(X1, X2, X3)
U131¹(X1, active(X2), X3) → U131¹(X1, X2, X3)
U131¹(X1, X2, active(X3)) → U131¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(471) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U131¹(X1, active(X2), X3) → U131¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131¹(x1, x2, x3)  =  U131¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U131^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(472) Obligation:

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

U131¹(mark(X1), X2, X3) → U131¹(X1, X2, X3)
U131¹(X1, X2, mark(X3)) → U131¹(X1, X2, X3)
U131¹(active(X1), X2, X3) → U131¹(X1, X2, X3)
U131¹(X1, X2, active(X3)) → U131¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(473) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U131¹(X1, X2, mark(X3)) → U131¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(474) Obligation:

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

U131¹(mark(X1), X2, X3) → U131¹(X1, X2, X3)
U131¹(active(X1), X2, X3) → U131¹(X1, X2, X3)
U131¹(X1, X2, active(X3)) → U131¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(475) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U131¹(active(X1), X2, X3) → U131¹(X1, X2, X3)
U131¹(X1, X2, active(X3)) → U131¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131¹(x1, x2, x3)  =  U131¹(x1, x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U131^1₂

Status:

active₁: [1]
U131^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(476) Obligation:

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

U131¹(mark(X1), X2, X3) → U131¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(477) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U131¹(mark(X1), X2, X3) → U131¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131¹(x1, x2, x3)  =  U131¹(x1, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U131^1₂

Status:

U131^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(478) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(479) PisEmptyProof (EQUIVALENT transformation)

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

(480) TRUE

(481) Obligation:

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

U122¹(active(X)) → U122¹(X)
U122¹(mark(X)) → U122¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(482) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U122¹(mark(X)) → U122¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U122¹(x1)  =  U122¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U122^1₁

Status:

U122^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(483) Obligation:

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

U122¹(active(X)) → U122¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(484) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U122¹(active(X)) → U122¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U122¹(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(485) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(486) PisEmptyProof (EQUIVALENT transformation)

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

(487) TRUE

(488) Obligation:

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

U121¹(X1, mark(X2)) → U121¹(X1, X2)
U121¹(mark(X1), X2) → U121¹(X1, X2)
U121¹(active(X1), X2) → U121¹(X1, X2)
U121¹(X1, active(X2)) → U121¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(489) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U121¹(X1, mark(X2)) → U121¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(490) Obligation:

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

U121¹(mark(X1), X2) → U121¹(X1, X2)
U121¹(active(X1), X2) → U121¹(X1, X2)
U121¹(X1, active(X2)) → U121¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(491) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U121¹(mark(X1), X2) → U121¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121¹(x1, x2)  =  U121¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U121^1₂

Status:

mark₁: [1]
U121^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(492) Obligation:

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

U121¹(active(X1), X2) → U121¹(X1, X2)
U121¹(X1, active(X2)) → U121¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(493) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U121¹(X1, active(X2)) → U121¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(494) Obligation:

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

U121¹(active(X1), X2) → U121¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(495) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U121¹(active(X1), X2) → U121¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(496) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(497) PisEmptyProof (EQUIVALENT transformation)

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

(498) TRUE

(499) Obligation:

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

U112¹(active(X)) → U112¹(X)
U112¹(mark(X)) → U112¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(500) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U112¹(mark(X)) → U112¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U112¹(x1)  =  U112¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U112^1₁

Status:

U112^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(501) Obligation:

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

U112¹(active(X)) → U112¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(502) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U112¹(active(X)) → U112¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U112¹(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(503) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(504) PisEmptyProof (EQUIVALENT transformation)

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

(505) TRUE

(506) Obligation:

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

U111¹(X1, mark(X2)) → U111¹(X1, X2)
U111¹(mark(X1), X2) → U111¹(X1, X2)
U111¹(active(X1), X2) → U111¹(X1, X2)
U111¹(X1, active(X2)) → U111¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(507) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U111¹(X1, mark(X2)) → U111¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(508) Obligation:

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

U111¹(mark(X1), X2) → U111¹(X1, X2)
U111¹(active(X1), X2) → U111¹(X1, X2)
U111¹(X1, active(X2)) → U111¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(509) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U111¹(mark(X1), X2) → U111¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111¹(x1, x2)  =  U111¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U111^1₂

Status:

mark₁: [1]
U111^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(510) Obligation:

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

U111¹(active(X1), X2) → U111¹(X1, X2)
U111¹(X1, active(X2)) → U111¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(511) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U111¹(X1, active(X2)) → U111¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(512) Obligation:

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

U111¹(active(X1), X2) → U111¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(513) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U111¹(active(X1), X2) → U111¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(514) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(515) PisEmptyProof (EQUIVALENT transformation)

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

(516) TRUE

(517) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(518) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(519) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(520) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > SPLITAT₂

Status:

SPLITAT₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(521) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(522) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(523) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(524) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(525) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(526) PisEmptyProof (EQUIVALENT transformation)

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

(527) TRUE

(528) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(529) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SND(mark(X)) → SND(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > SND₁

Status:

mark₁: [1]
SND₁: [1]

The following usable rules [FROCOS05] were oriented: none

(530) Obligation:

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

SND(active(X)) → SND(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(531) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SND(active(X)) → SND(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(532) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(533) PisEmptyProof (EQUIVALENT transformation)

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

(534) TRUE

(535) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(536) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U11^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(537) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(538) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U11^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(539) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(540) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(541) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(542) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Precedence:

active₁ > U11^1₂

Status:

active₁: [1]
U11^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(543) 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, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(544) 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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U11^1₂

Status:

U11^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(545) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(546) PisEmptyProof (EQUIVALENT transformation)

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

(547) TRUE

(548) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(549) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISLNAT(mark(X)) → ISLNAT(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > ISLNAT₁

Status:

ISLNAT₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(550) Obligation:

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

ISLNAT(active(X)) → ISLNAT(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(551) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISLNAT(active(X)) → ISLNAT(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(552) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(553) PisEmptyProof (EQUIVALENT transformation)

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

(554) TRUE

(555) Obligation:

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

U103¹(active(X)) → U103¹(X)
U103¹(mark(X)) → U103¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(556) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U103¹(mark(X)) → U103¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U103¹(x1)  =  U103¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U103^1₁

Status:

U103^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(557) Obligation:

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

U103¹(active(X)) → U103¹(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(558) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U103¹(active(X)) → U103¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U103¹(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(559) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(560) PisEmptyProof (EQUIVALENT transformation)

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

(561) TRUE

(562) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(563) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATURAL(mark(X)) → ISNATURAL(X)

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

Lexicographic path order with status [LPO].
Precedence:

mark₁ > ISNATURAL₁

Status:

ISNATURAL₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(564) Obligation:

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

ISNATURAL(active(X)) → ISNATURAL(X)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(565) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATURAL(active(X)) → ISNATURAL(X)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(566) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(567) PisEmptyProof (EQUIVALENT transformation)

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

(568) TRUE

(569) Obligation:

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

U102¹(X1, mark(X2)) → U102¹(X1, X2)
U102¹(mark(X1), X2) → U102¹(X1, X2)
U102¹(active(X1), X2) → U102¹(X1, X2)
U102¹(X1, active(X2)) → U102¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(570) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(X1, mark(X2)) → U102¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(571) Obligation:

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

U102¹(mark(X1), X2) → U102¹(X1, X2)
U102¹(active(X1), X2) → U102¹(X1, X2)
U102¹(X1, active(X2)) → U102¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(572) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(mark(X1), X2) → U102¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2)  =  U102¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U102^1₂

Status:

U102^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(573) Obligation:

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

U102¹(active(X1), X2) → U102¹(X1, X2)
U102¹(X1, active(X2)) → U102¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(574) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(X1, active(X2)) → U102¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(575) Obligation:

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

U102¹(active(X1), X2) → U102¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(576) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(active(X1), X2) → U102¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(577) Obligation:

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

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(578) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(579) TRUE

(580) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U101¹(X1, mark(X2), X3) → U101¹(X1, X2, X3)
U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, mark(X3)) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, active(X2), X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(581) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(X1, mark(X2), X3) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  U101¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U101^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(582) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, mark(X3)) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, active(X2), X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(583) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(X1, active(X2), X3) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  U101¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U101^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(584) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, mark(X3)) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(585) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(X1, X2, mark(X3)) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(586) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(587) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  U101¹(x1, x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U101^1₂

Status:

active₁: [1]
U101^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(588) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(589) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  U101¹(x1, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U101^1₂

Status:

mark₁: [1]
U101^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(590) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(591) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(592) TRUE

(593) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U101(X1, X2, X3)) → ACTIVE(U101(mark(X1), X2, X3))
ACTIVE(U101(tt, V1, V2)) → MARK(U102(isNatural(V1), V2))
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2)) → ACTIVE(U102(mark(X1), X2))
ACTIVE(U102(tt, V2)) → MARK(U103(isLNat(V2)))
MARK(U102(X1, X2)) → MARK(X1)
MARK(isNatural(X)) → ACTIVE(isNatural(X))
ACTIVE(U11(tt, N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(U103(X)) → ACTIVE(U103(mark(X)))
ACTIVE(U111(tt, V1)) → MARK(U112(isLNat(V1)))
MARK(U103(X)) → MARK(X)
MARK(isLNat(X)) → ACTIVE(isLNat(X))
ACTIVE(U121(tt, V1)) → MARK(U122(isNatural(V1)))
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U131(tt, V1, V2)) → MARK(U132(isNatural(V1), V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
ACTIVE(U132(tt, V2)) → MARK(U133(isLNat(V2)))
MARK(snd(X)) → MARK(X)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
ACTIVE(U141(tt, V1, V2)) → MARK(U142(isLNat(V1), V2))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U111(X1, X2)) → ACTIVE(U111(mark(X1), X2))
ACTIVE(U142(tt, V2)) → MARK(U143(isLNat(V2)))
MARK(U111(X1, X2)) → MARK(X1)
MARK(U112(X)) → ACTIVE(U112(mark(X)))
ACTIVE(U151(tt, V1, V2)) → MARK(U152(isNatural(V1), V2))
MARK(U112(X)) → MARK(X)
MARK(U121(X1, X2)) → ACTIVE(U121(mark(X1), X2))
ACTIVE(U152(tt, V2)) → MARK(U153(isLNat(V2)))
MARK(U121(X1, X2)) → MARK(X1)
MARK(U122(X)) → ACTIVE(U122(mark(X)))
ACTIVE(U161(tt, N)) → MARK(cons(N, natsFrom(s(N))))
MARK(U122(X)) → MARK(X)
MARK(U131(X1, X2, X3)) → ACTIVE(U131(mark(X1), X2, X3))
ACTIVE(U171(tt, N, XS)) → MARK(head(afterNth(N, XS)))
MARK(U131(X1, X2, X3)) → MARK(X1)
MARK(U132(X1, X2)) → ACTIVE(U132(mark(X1), X2))
ACTIVE(U181(tt, Y)) → MARK(Y)
MARK(U132(X1, X2)) → MARK(X1)
MARK(U133(X)) → ACTIVE(U133(mark(X)))
ACTIVE(U191(tt, XS)) → MARK(pair(nil, XS))
MARK(U133(X)) → MARK(X)
MARK(U141(X1, X2, X3)) → ACTIVE(U141(mark(X1), X2, X3))
ACTIVE(U201(tt, N, X, XS)) → MARK(U202(splitAt(N, XS), X))
MARK(U141(X1, X2, X3)) → MARK(X1)
MARK(U142(X1, X2)) → ACTIVE(U142(mark(X1), X2))
ACTIVE(U202(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(U142(X1, X2)) → MARK(X1)
MARK(U143(X)) → ACTIVE(U143(mark(X)))
ACTIVE(U21(tt, X)) → MARK(X)
MARK(U143(X)) → MARK(X)
MARK(U151(X1, X2, X3)) → ACTIVE(U151(mark(X1), X2, X3))
ACTIVE(U211(tt, XS)) → MARK(XS)
MARK(U151(X1, X2, X3)) → MARK(X1)
MARK(U152(X1, X2)) → ACTIVE(U152(mark(X1), X2))
ACTIVE(U221(tt, N, XS)) → MARK(fst(splitAt(N, XS)))
MARK(U152(X1, X2)) → MARK(X1)
MARK(U153(X)) → ACTIVE(U153(mark(X)))
ACTIVE(U31(tt, N)) → MARK(N)
MARK(U153(X)) → MARK(X)
MARK(U161(X1, X2)) → ACTIVE(U161(mark(X1), X2))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isNatural(V1), V2))
MARK(U161(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U42(tt, V2)) → MARK(U43(isLNat(V2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNatural(V1), V2))
MARK(natsFrom(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(U52(tt, V2)) → MARK(U53(isLNat(V2)))
MARK(s(X)) → MARK(X)
MARK(U171(X1, X2, X3)) → ACTIVE(U171(mark(X1), X2, X3))
ACTIVE(U61(tt, V1)) → MARK(U62(isPLNat(V1)))
MARK(U171(X1, X2, X3)) → MARK(X1)
MARK(head(X)) → ACTIVE(head(mark(X)))
ACTIVE(U71(tt, V1)) → MARK(U72(isNatural(V1)))
MARK(head(X)) → MARK(X)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
ACTIVE(U81(tt, V1)) → MARK(U82(isPLNat(V1)))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(U181(X1, X2)) → ACTIVE(U181(mark(X1), X2))
ACTIVE(U91(tt, V1)) → MARK(U92(isLNat(V1)))
MARK(U181(X1, X2)) → MARK(X1)
MARK(U191(X1, X2)) → ACTIVE(U191(mark(X1), X2))
ACTIVE(afterNth(N, XS)) → MARK(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
MARK(U191(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → ACTIVE(pair(mark(X1), mark(X2)))
ACTIVE(and(tt, X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
MARK(U201(X1, X2, X3, X4)) → ACTIVE(U201(mark(X1), X2, X3, X4))
ACTIVE(fst(pair(X, Y))) → MARK(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
MARK(U201(X1, X2, X3, X4)) → MARK(X1)
MARK(U202(X1, X2)) → ACTIVE(U202(mark(X1), X2))
ACTIVE(head(cons(N, XS))) → MARK(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
MARK(U202(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
ACTIVE(isLNat(afterNth(V1, V2))) → MARK(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U211(X1, X2)) → ACTIVE(U211(mark(X1), X2))
ACTIVE(isLNat(cons(V1, V2))) → MARK(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
MARK(U211(X1, X2)) → MARK(X1)
MARK(U221(X1, X2, X3)) → ACTIVE(U221(mark(X1), X2, X3))
ACTIVE(isLNat(fst(V1))) → MARK(U61(isPLNatKind(V1), V1))
MARK(U221(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
ACTIVE(isLNat(natsFrom(V1))) → MARK(U71(isNaturalKind(V1), V1))
MARK(fst(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(isLNat(snd(V1))) → MARK(U81(isPLNatKind(V1), V1))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
ACTIVE(isLNat(tail(V1))) → MARK(U91(isLNatKind(V1), V1))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(isLNat(take(V1, V2))) → MARK(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
MARK(U42(X1, X2)) → MARK(X1)
MARK(U43(X)) → ACTIVE(U43(mark(X)))
ACTIVE(isLNatKind(afterNth(V1, V2))) → MARK(and(isNaturalKind(V1), isLNatKind(V2)))
MARK(U43(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
ACTIVE(isLNatKind(cons(V1, V2))) → MARK(and(isNaturalKind(V1), isLNatKind(V2)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(isLNatKind(fst(V1))) → MARK(isPLNatKind(V1))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U53(X)) → ACTIVE(U53(mark(X)))
ACTIVE(isLNatKind(natsFrom(V1))) → MARK(isNaturalKind(V1))
MARK(U53(X)) → MARK(X)
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isLNatKind(snd(V1))) → MARK(isPLNatKind(V1))
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → ACTIVE(U62(mark(X)))
ACTIVE(isLNatKind(tail(V1))) → MARK(isLNatKind(V1))
MARK(U62(X)) → MARK(X)
MARK(isPLNat(X)) → ACTIVE(isPLNat(X))
ACTIVE(isLNatKind(take(V1, V2))) → MARK(and(isNaturalKind(V1), isLNatKind(V2)))
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(isNatural(head(V1))) → MARK(U111(isLNatKind(V1), V1))
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → ACTIVE(U72(mark(X)))
ACTIVE(isNatural(s(V1))) → MARK(U121(isNaturalKind(V1), V1))
MARK(U72(X)) → MARK(X)
MARK(U81(X1, X2)) → ACTIVE(U81(mark(X1), X2))
ACTIVE(isNatural(sel(V1, V2))) → MARK(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
MARK(U81(X1, X2)) → MARK(X1)
MARK(U82(X)) → ACTIVE(U82(mark(X)))
ACTIVE(isNaturalKind(head(V1))) → MARK(isLNatKind(V1))
MARK(U82(X)) → MARK(X)
MARK(U91(X1, X2)) → ACTIVE(U91(mark(X1), X2))
ACTIVE(isNaturalKind(s(V1))) → MARK(isNaturalKind(V1))
MARK(U91(X1, X2)) → MARK(X1)
MARK(U92(X)) → ACTIVE(U92(mark(X)))
ACTIVE(isNaturalKind(sel(V1, V2))) → MARK(and(isNaturalKind(V1), isLNatKind(V2)))
MARK(U92(X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isPLNat(pair(V1, V2))) → MARK(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNaturalKind(X)) → ACTIVE(isNaturalKind(X))
ACTIVE(isPLNat(splitAt(V1, V2))) → MARK(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
MARK(isLNatKind(X)) → ACTIVE(isLNatKind(X))
ACTIVE(isPLNatKind(pair(V1, V2))) → MARK(and(isLNatKind(V1), isLNatKind(V2)))
MARK(isPLNatKind(X)) → ACTIVE(isPLNatKind(X))
ACTIVE(isPLNatKind(splitAt(V1, V2))) → MARK(and(isNaturalKind(V1), isLNatKind(V2)))
MARK(tail(X)) → ACTIVE(tail(mark(X)))
ACTIVE(natsFrom(N)) → MARK(U161(and(isNatural(N), isNaturalKind(N)), N))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(sel(N, XS)) → MARK(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(snd(pair(X, Y))) → MARK(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
ACTIVE(splitAt(0, XS)) → MARK(U191(and(isLNat(XS), isLNatKind(XS)), XS))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
ACTIVE(tail(cons(N, XS))) → MARK(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
ACTIVE(take(N, XS)) → MARK(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))

The TRS R consists of the following rules:

active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2)) → active(U102(mark(X1), X2))
mark(isNatural(X)) → active(isNatural(X))
mark(U103(X)) → active(U103(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U111(X1, X2)) → active(U111(mark(X1), X2))
mark(U112(X)) → active(U112(mark(X)))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(U131(X1, X2, X3)) → active(U131(mark(X1), X2, X3))
mark(U132(X1, X2)) → active(U132(mark(X1), X2))
mark(U133(X)) → active(U133(mark(X)))
mark(U141(X1, X2, X3)) → active(U141(mark(X1), X2, X3))
mark(U142(X1, X2)) → active(U142(mark(X1), X2))
mark(U143(X)) → active(U143(mark(X)))
mark(U151(X1, X2, X3)) → active(U151(mark(X1), X2, X3))
mark(U152(X1, X2)) → active(U152(mark(X1), X2))
mark(U153(X)) → active(U153(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2)) → active(U202(mark(X1), X2))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X)) → active(U72(mark(X)))
mark(U81(X1, X2)) → active(U81(mark(X1), X2))
mark(U82(X)) → active(U82(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNaturalKind(X)) → active(isNaturalKind(X))
mark(isLNatKind(X)) → active(isLNatKind(X))
mark(isPLNatKind(X)) → active(isPLNatKind(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2) → U102(X1, X2)
U102(X1, mark(X2)) → U102(X1, X2)
U102(active(X1), X2) → U102(X1, X2)
U102(X1, active(X2)) → U102(X1, X2)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U103(mark(X)) → U103(X)
U103(active(X)) → U103(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U112(mark(X)) → U112(X)
U112(active(X)) → U112(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
U132(mark(X1), X2) → U132(X1, X2)
U132(X1, mark(X2)) → U132(X1, X2)
U132(active(X1), X2) → U132(X1, X2)
U132(X1, active(X2)) → U132(X1, X2)
U133(mark(X)) → U133(X)
U133(active(X)) → U133(X)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U142(mark(X1), X2) → U142(X1, X2)
U142(X1, mark(X2)) → U142(X1, X2)
U142(active(X1), X2) → U142(X1, X2)
U142(X1, active(X2)) → U142(X1, X2)
U143(mark(X)) → U143(X)
U143(active(X)) → U143(X)
U151(mark(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, mark(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, mark(X3)) → U151(X1, X2, X3)
U151(active(X1), X2, X3) → U151(X1, X2, X3)
U151(X1, active(X2), X3) → U151(X1, X2, X3)
U151(X1, X2, active(X3)) → U151(X1, X2, X3)
U152(mark(X1), X2) → U152(X1, X2)
U152(X1, mark(X2)) → U152(X1, X2)
U152(active(X1), X2) → U152(X1, X2)
U152(X1, active(X2)) → U152(X1, X2)
U153(mark(X)) → U153(X)
U153(active(X)) → U153(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2) → U202(X1, X2)
U202(X1, mark(X2)) → U202(X1, X2)
U202(active(X1), X2) → U202(X1, X2)
U202(X1, active(X2)) → U202(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X)) → U72(X)
U72(active(X)) → U72(X)
U81(mark(X1), X2) → U81(X1, X2)
U81(X1, mark(X2)) → U81(X1, X2)
U81(active(X1), X2) → U81(X1, X2)
U81(X1, active(X2)) → U81(X1, X2)
U82(mark(X)) → U82(X)
U82(active(X)) → U82(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNaturalKind(mark(X)) → isNaturalKind(X)
isNaturalKind(active(X)) → isNaturalKind(X)
isLNatKind(mark(X)) → isLNatKind(X)
isLNatKind(active(X)) → isLNatKind(X)
isPLNatKind(mark(X)) → isPLNatKind(X)
isPLNatKind(active(X)) → isPLNatKind(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.