Problem Transformed CSR 04 LISTUTILITIES nokinds noand C

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 LISTUTILITIES nokinds noand C

stdout:

MAYBE

Problem:
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))

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 LISTUTILITIES nokinds noand C

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	TIMEOUT
Input	Transformed CSR 04 LISTUTILITIES nokinds noand C

stdout:

TIMEOUT

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           TIMEOUT
Input Problem:    innermost runtime-complexity with respect to
  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))}

Proof Output:
  Computation stopped due to timeout after 60.0 seconds

Tool RC2

Execution Time	Unknown
Answer	TIMEOUT
Input	Transformed CSR 04 LISTUTILITIES nokinds noand C

stdout:

TIMEOUT

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           TIMEOUT
Input Problem:    runtime-complexity with respect to
  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))}

Proof Output:
  Computation stopped due to timeout after 60.0 seconds