Problem Transformed CSR 04 LISTUTILITIES complete C

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 LISTUTILITIES complete C

stdout:

MAYBE

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

Proof:
Open

Tool IRC2

Execution Time	Unknown
Answer	TIMEOUT
Input	Transformed CSR 04 LISTUTILITIES complete C

stdout:

TIMEOUT

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           TIMEOUT
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  active(U101(tt(), V1, V2)) -> mark(U102(isNatural(V1), V2))
     , active(U102(tt(), V2)) -> mark(U103(isLNat(V2)))
     , active(U103(tt())) -> mark(tt())
     , active(U11(tt(), N, XS)) -> mark(snd(splitAt(N, XS)))
     , active(U111(tt(), V1)) -> mark(U112(isLNat(V1)))
     , active(U112(tt())) -> mark(tt())
     , active(U121(tt(), V1)) -> mark(U122(isNatural(V1)))
     , active(U122(tt())) -> mark(tt())
     , active(U131(tt(), V1, V2)) -> mark(U132(isNatural(V1), V2))
     , active(U132(tt(), V2)) -> mark(U133(isLNat(V2)))
     , active(U133(tt())) -> mark(tt())
     , active(U141(tt(), V1, V2)) -> mark(U142(isLNat(V1), V2))
     , active(U142(tt(), V2)) -> mark(U143(isLNat(V2)))
     , active(U143(tt())) -> mark(tt())
     , active(U151(tt(), V1, V2)) -> mark(U152(isNatural(V1), V2))
     , active(U152(tt(), V2)) -> mark(U153(isLNat(V2)))
     , active(U153(tt())) -> mark(tt())
     , active(U161(tt(), N)) -> mark(cons(N, natsFrom(s(N))))
     , active(U171(tt(), N, XS)) -> mark(head(afterNth(N, XS)))
     , active(U181(tt(), Y)) -> mark(Y)
     , active(U191(tt(), XS)) -> mark(pair(nil(), XS))
     , active(U201(tt(), N, X, XS)) -> mark(U202(splitAt(N, XS), X))
     , active(U202(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
     , active(U21(tt(), X)) -> mark(X)
     , active(U211(tt(), XS)) -> mark(XS)
     , active(U221(tt(), N, XS)) -> mark(fst(splitAt(N, XS)))
     , active(U31(tt(), N)) -> mark(N)
     , active(U41(tt(), V1, V2)) -> mark(U42(isNatural(V1), V2))
     , active(U42(tt(), V2)) -> mark(U43(isLNat(V2)))
     , active(U43(tt())) -> mark(tt())
     , active(U51(tt(), V1, V2)) -> mark(U52(isNatural(V1), V2))
     , active(U52(tt(), V2)) -> mark(U53(isLNat(V2)))
     , active(U53(tt())) -> mark(tt())
     , active(U61(tt(), V1)) -> mark(U62(isPLNat(V1)))
     , active(U62(tt())) -> mark(tt())
     , active(U71(tt(), V1)) -> mark(U72(isNatural(V1)))
     , active(U72(tt())) -> mark(tt())
     , active(U81(tt(), V1)) -> mark(U82(isPLNat(V1)))
     , active(U82(tt())) -> mark(tt())
     , active(U91(tt(), V1)) -> mark(U92(isLNat(V1)))
     , active(U92(tt())) -> mark(tt())
     , active(afterNth(N, XS)) ->
       mark(U11(and(and(isNatural(N), isNaturalKind(N)),
                    and(isLNat(XS), isLNatKind(XS))),
                N,
                XS))
     , active(and(tt(), X)) -> mark(X)
     , active(fst(pair(X, Y))) ->
       mark(U21(and(and(isLNat(X), isLNatKind(X)),
                    and(isLNat(Y), isLNatKind(Y))),
                X))
     , active(head(cons(N, XS))) ->
       mark(U31(and(and(isNatural(N), isNaturalKind(N)),
                    and(isLNat(XS), isLNatKind(XS))),
                N))
     , active(isLNat(nil())) -> mark(tt())
     , active(isLNat(afterNth(V1, V2))) ->
       mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
     , active(isLNat(cons(V1, V2))) ->
       mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
     , active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1), V1))
     , active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1), V1))
     , active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1), V1))
     , active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1), V1))
     , active(isLNat(take(V1, V2))) ->
       mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
     , active(isLNatKind(nil())) -> mark(tt())
     , active(isLNatKind(afterNth(V1, V2))) ->
       mark(and(isNaturalKind(V1), isLNatKind(V2)))
     , active(isLNatKind(cons(V1, V2))) ->
       mark(and(isNaturalKind(V1), isLNatKind(V2)))
     , active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
     , active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
     , active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
     , active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
     , active(isLNatKind(take(V1, V2))) ->
       mark(and(isNaturalKind(V1), isLNatKind(V2)))
     , active(isNatural(0())) -> mark(tt())
     , active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1), V1))
     , active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1), V1))
     , active(isNatural(sel(V1, V2))) ->
       mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
     , active(isNaturalKind(0())) -> mark(tt())
     , active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
     , active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
     , active(isNaturalKind(sel(V1, V2))) ->
       mark(and(isNaturalKind(V1), isLNatKind(V2)))
     , active(isPLNat(pair(V1, V2))) ->
       mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
     , active(isPLNat(splitAt(V1, V2))) ->
       mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
     , active(isPLNatKind(pair(V1, V2))) ->
       mark(and(isLNatKind(V1), isLNatKind(V2)))
     , active(isPLNatKind(splitAt(V1, V2))) ->
       mark(and(isNaturalKind(V1), isLNatKind(V2)))
     , active(natsFrom(N)) ->
       mark(U161(and(isNatural(N), isNaturalKind(N)), N))
     , active(sel(N, XS)) ->
       mark(U171(and(and(isNatural(N), isNaturalKind(N)),
                     and(isLNat(XS), isLNatKind(XS))),
                 N,
                 XS))
     , active(snd(pair(X, Y))) ->
       mark(U181(and(and(isLNat(X), isLNatKind(X)),
                     and(isLNat(Y), isLNatKind(Y))),
                 Y))
     , active(splitAt(0(), XS)) ->
       mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
     , active(splitAt(s(N), cons(X, XS))) ->
       mark(U201(and(and(isNatural(N), isNaturalKind(N)),
                     and(and(isNatural(X), isNaturalKind(X)),
                         and(isLNat(XS), isLNatKind(XS)))),
                 N,
                 X,
                 XS))
     , active(tail(cons(N, XS))) ->
       mark(U211(and(and(isNatural(N), isNaturalKind(N)),
                     and(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))}

Proof Output:
  Computation stopped due to timeout after 60.0 seconds

Tool RC2

Execution Time	Unknown
Answer	TIMEOUT
Input	Transformed CSR 04 LISTUTILITIES complete C

stdout:

TIMEOUT

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           TIMEOUT
Input Problem:    runtime-complexity with respect to
  Rules:
    {  active(U101(tt(), V1, V2)) -> mark(U102(isNatural(V1), V2))
     , active(U102(tt(), V2)) -> mark(U103(isLNat(V2)))
     , active(U103(tt())) -> mark(tt())
     , active(U11(tt(), N, XS)) -> mark(snd(splitAt(N, XS)))
     , active(U111(tt(), V1)) -> mark(U112(isLNat(V1)))
     , active(U112(tt())) -> mark(tt())
     , active(U121(tt(), V1)) -> mark(U122(isNatural(V1)))
     , active(U122(tt())) -> mark(tt())
     , active(U131(tt(), V1, V2)) -> mark(U132(isNatural(V1), V2))
     , active(U132(tt(), V2)) -> mark(U133(isLNat(V2)))
     , active(U133(tt())) -> mark(tt())
     , active(U141(tt(), V1, V2)) -> mark(U142(isLNat(V1), V2))
     , active(U142(tt(), V2)) -> mark(U143(isLNat(V2)))
     , active(U143(tt())) -> mark(tt())
     , active(U151(tt(), V1, V2)) -> mark(U152(isNatural(V1), V2))
     , active(U152(tt(), V2)) -> mark(U153(isLNat(V2)))
     , active(U153(tt())) -> mark(tt())
     , active(U161(tt(), N)) -> mark(cons(N, natsFrom(s(N))))
     , active(U171(tt(), N, XS)) -> mark(head(afterNth(N, XS)))
     , active(U181(tt(), Y)) -> mark(Y)
     , active(U191(tt(), XS)) -> mark(pair(nil(), XS))
     , active(U201(tt(), N, X, XS)) -> mark(U202(splitAt(N, XS), X))
     , active(U202(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
     , active(U21(tt(), X)) -> mark(X)
     , active(U211(tt(), XS)) -> mark(XS)
     , active(U221(tt(), N, XS)) -> mark(fst(splitAt(N, XS)))
     , active(U31(tt(), N)) -> mark(N)
     , active(U41(tt(), V1, V2)) -> mark(U42(isNatural(V1), V2))
     , active(U42(tt(), V2)) -> mark(U43(isLNat(V2)))
     , active(U43(tt())) -> mark(tt())
     , active(U51(tt(), V1, V2)) -> mark(U52(isNatural(V1), V2))
     , active(U52(tt(), V2)) -> mark(U53(isLNat(V2)))
     , active(U53(tt())) -> mark(tt())
     , active(U61(tt(), V1)) -> mark(U62(isPLNat(V1)))
     , active(U62(tt())) -> mark(tt())
     , active(U71(tt(), V1)) -> mark(U72(isNatural(V1)))
     , active(U72(tt())) -> mark(tt())
     , active(U81(tt(), V1)) -> mark(U82(isPLNat(V1)))
     , active(U82(tt())) -> mark(tt())
     , active(U91(tt(), V1)) -> mark(U92(isLNat(V1)))
     , active(U92(tt())) -> mark(tt())
     , active(afterNth(N, XS)) ->
       mark(U11(and(and(isNatural(N), isNaturalKind(N)),
                    and(isLNat(XS), isLNatKind(XS))),
                N,
                XS))
     , active(and(tt(), X)) -> mark(X)
     , active(fst(pair(X, Y))) ->
       mark(U21(and(and(isLNat(X), isLNatKind(X)),
                    and(isLNat(Y), isLNatKind(Y))),
                X))
     , active(head(cons(N, XS))) ->
       mark(U31(and(and(isNatural(N), isNaturalKind(N)),
                    and(isLNat(XS), isLNatKind(XS))),
                N))
     , active(isLNat(nil())) -> mark(tt())
     , active(isLNat(afterNth(V1, V2))) ->
       mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
     , active(isLNat(cons(V1, V2))) ->
       mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
     , active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1), V1))
     , active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1), V1))
     , active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1), V1))
     , active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1), V1))
     , active(isLNat(take(V1, V2))) ->
       mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
     , active(isLNatKind(nil())) -> mark(tt())
     , active(isLNatKind(afterNth(V1, V2))) ->
       mark(and(isNaturalKind(V1), isLNatKind(V2)))
     , active(isLNatKind(cons(V1, V2))) ->
       mark(and(isNaturalKind(V1), isLNatKind(V2)))
     , active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
     , active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
     , active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
     , active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
     , active(isLNatKind(take(V1, V2))) ->
       mark(and(isNaturalKind(V1), isLNatKind(V2)))
     , active(isNatural(0())) -> mark(tt())
     , active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1), V1))
     , active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1), V1))
     , active(isNatural(sel(V1, V2))) ->
       mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
     , active(isNaturalKind(0())) -> mark(tt())
     , active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
     , active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
     , active(isNaturalKind(sel(V1, V2))) ->
       mark(and(isNaturalKind(V1), isLNatKind(V2)))
     , active(isPLNat(pair(V1, V2))) ->
       mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
     , active(isPLNat(splitAt(V1, V2))) ->
       mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
     , active(isPLNatKind(pair(V1, V2))) ->
       mark(and(isLNatKind(V1), isLNatKind(V2)))
     , active(isPLNatKind(splitAt(V1, V2))) ->
       mark(and(isNaturalKind(V1), isLNatKind(V2)))
     , active(natsFrom(N)) ->
       mark(U161(and(isNatural(N), isNaturalKind(N)), N))
     , active(sel(N, XS)) ->
       mark(U171(and(and(isNatural(N), isNaturalKind(N)),
                     and(isLNat(XS), isLNatKind(XS))),
                 N,
                 XS))
     , active(snd(pair(X, Y))) ->
       mark(U181(and(and(isLNat(X), isLNatKind(X)),
                     and(isLNat(Y), isLNatKind(Y))),
                 Y))
     , active(splitAt(0(), XS)) ->
       mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
     , active(splitAt(s(N), cons(X, XS))) ->
       mark(U201(and(and(isNatural(N), isNaturalKind(N)),
                     and(and(isNatural(X), isNaturalKind(X)),
                         and(isLNat(XS), isLNatKind(XS)))),
                 N,
                 X,
                 XS))
     , active(tail(cons(N, XS))) ->
       mark(U211(and(and(isNatural(N), isNaturalKind(N)),
                     and(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))}

Proof Output:
  Computation stopped due to timeout after 60.0 seconds