Termination w.r.t. Q proof of /tmp/tmpARgpeK/LISTUTILITIES_complete

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

0 QTRS

(0) Obligation:

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

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

Q is empty.