The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

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

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
U101(mark(X1), X2) →⁺ mark(U101(X1, X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
active, U102, isLNat, U12, snd, splitAt, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U102 < active
isLNat < active
U12 < active
snd < active
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U102 < proper
isLNat < proper
U12 < proper
snd < proper
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(8) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U102, active, isLNat, U12, snd, splitAt, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U102 < active
isLNat < active
U12 < active
snd < active
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U102 < proper
isLNat < proper
U12 < proper
snd < proper
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U102.

(10) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
isLNat, active, U12, snd, splitAt, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
isLNat < active
U12 < active
snd < active
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isLNat < proper
U12 < proper
snd < proper
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol isLNat.

(12) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U12, active, snd, splitAt, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U12 < active
snd < active
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U12 < proper
snd < proper
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U12.

(14) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
snd, active, splitAt, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
snd < active
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
snd < proper
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol snd.

(16) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
splitAt, active, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol splitAt.

(18) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U132, active, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U132.

(20) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U142, active, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U142.

(22) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U152, active, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U152.

(24) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
cons, active, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol cons.

(26) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
natsFrom, active, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol natsFrom.

(28) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
s, active, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(29) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol s.

(30) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U172, active, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(31) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U172.

(32) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
head, active, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(33) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol head.

(34) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
afterNth, active, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(35) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol afterNth.

(36) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U182, active, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(37) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U182.

(38) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
pair, active, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(39) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol pair.

(40) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U202, active, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(41) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U202.

(42) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
isNatural, active, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(43) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol isNatural.

(44) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U203, active, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(45) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U203.

(46) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U204, active, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(47) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U204.

(48) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U22, active, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(49) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U22.

(50) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U212, active, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(51) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U212.

(52) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U222, active, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(53) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U222.

(54) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
fst, active, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(55) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol fst.

(56) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U32, active, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(57) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U32.

(58) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U42, active, U52, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(59) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U42.

(60) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U52, active, U11, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(61) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U52.

(62) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U11, active, U21, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(63) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U11.

(64) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U21, active, U31, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(65) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U21.

(66) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U31, active, U41, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(67) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U31.

(68) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U41, active, U51, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U41 < active
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U41 < proper
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(69) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U41.

(70) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U51, active, U61, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U51 < active
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U51 < proper
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(71) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U51.

(72) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U61, active, isPLNat, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U61 < active
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U61 < proper
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(73) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U61.

(74) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
isPLNat, active, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
isPLNat < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isPLNat < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(75) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol isPLNat.

(76) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U71, active, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(77) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U71.

(78) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U81, active, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(79) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U81.

(80) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U91, active, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(81) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U91.

(82) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U101, active, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(83) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U101.

(84) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U111, active, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(85) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U111.

(86) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U121, active, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(87) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U121.

(88) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U131, active, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(89) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U131.

(90) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U141, active, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(91) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U141.

(92) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U151, active, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(93) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U151.

(94) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U161, active, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(95) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U161.

(96) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U171, active, U181, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(97) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U171.

(98) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U181, active, U191, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(99) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U181.

(100) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U191, active, U201, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(101) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U191.

(102) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U201, active, U211, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(103) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U201.

(104) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U211, active, U221, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(105) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U211.

(106) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
U221, active, tail, take, sel, proper, top

They will be analysed ascendingly in the following order:
U221 < active
tail < active
take < active
sel < active
active < top
U221 < proper
tail < proper
take < proper
sel < proper
proper < top

(107) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U221.

(108) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
tail, active, take, sel, proper, top

They will be analysed ascendingly in the following order:
tail < active
take < active
sel < active
active < top
tail < proper
take < proper
sel < proper
proper < top

(109) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol tail.

(110) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
take, active, sel, proper, top

They will be analysed ascendingly in the following order:
take < active
sel < active
active < top
take < proper
sel < proper
proper < top

(111) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol take.

(112) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
sel, active, proper, top

They will be analysed ascendingly in the following order:
sel < active
active < top
sel < proper
proper < top

(113) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol sel.

(114) Obligation:

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

Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U12 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U172 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U182 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U204 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U22 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U212 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U222 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U32 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok

Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))

The following defined symbols remain to be analysed:
active, proper, top

They will be analysed ascendingly in the following order:
active < top
proper < top