Problem Transformed CSR 04 LISTUTILITIES nokinds noand GM

Tool CaT

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

stdout:

MAYBE

Problem:
a__U101(tt(),V2) -> a__U102(a__isLNat(V2))
a__U102(tt()) -> tt()
a__U11(tt(),N,XS) -> a__U12(a__isLNat(XS),N,XS)
a__U111(tt()) -> tt()
a__U12(tt(),N,XS) -> a__snd(a__splitAt(mark(N),mark(XS)))
a__U121(tt()) -> tt()
a__U131(tt(),V2) -> a__U132(a__isLNat(V2))
a__U132(tt()) -> tt()
a__U141(tt(),V2) -> a__U142(a__isLNat(V2))
a__U142(tt()) -> tt()
a__U151(tt(),V2) -> a__U152(a__isLNat(V2))
a__U152(tt()) -> tt()
a__U161(tt(),N) -> cons(mark(N),natsFrom(s(N)))
a__U171(tt(),N,XS) -> a__U172(a__isLNat(XS),N,XS)
a__U172(tt(),N,XS) -> a__head(a__afterNth(mark(N),mark(XS)))
a__U181(tt(),Y) -> a__U182(a__isLNat(Y),Y)
a__U182(tt(),Y) -> mark(Y)
a__U191(tt(),XS) -> pair(nil(),mark(XS))
a__U201(tt(),N,X,XS) -> a__U202(a__isNatural(X),N,X,XS)
a__U202(tt(),N,X,XS) -> a__U203(a__isLNat(XS),N,X,XS)
a__U203(tt(),N,X,XS) -> a__U204(a__splitAt(mark(N),mark(XS)),X)
a__U204(pair(YS,ZS),X) -> pair(cons(mark(X),YS),mark(ZS))
a__U21(tt(),X,Y) -> a__U22(a__isLNat(Y),X)
a__U211(tt(),XS) -> a__U212(a__isLNat(XS),XS)
a__U212(tt(),XS) -> mark(XS)
a__U22(tt(),X) -> mark(X)
a__U221(tt(),N,XS) -> a__U222(a__isLNat(XS),N,XS)
a__U222(tt(),N,XS) -> a__fst(a__splitAt(mark(N),mark(XS)))
a__U31(tt(),N,XS) -> a__U32(a__isLNat(XS),N)
a__U32(tt(),N) -> mark(N)
a__U41(tt(),V2) -> a__U42(a__isLNat(V2))
a__U42(tt()) -> tt()
a__U51(tt(),V2) -> a__U52(a__isLNat(V2))
a__U52(tt()) -> tt()
a__U61(tt()) -> tt()
a__U71(tt()) -> tt()
a__U81(tt()) -> tt()
a__U91(tt()) -> tt()
a__afterNth(N,XS) -> a__U11(a__isNatural(N),N,XS)
a__fst(pair(X,Y)) -> a__U21(a__isLNat(X),X,Y)
a__head(cons(N,XS)) -> a__U31(a__isNatural(N),N,XS)
a__isLNat(nil()) -> tt()
a__isLNat(afterNth(V1,V2)) -> a__U41(a__isNatural(V1),V2)
a__isLNat(cons(V1,V2)) -> a__U51(a__isNatural(V1),V2)
a__isLNat(fst(V1)) -> a__U61(a__isPLNat(V1))
a__isLNat(natsFrom(V1)) -> a__U71(a__isNatural(V1))
a__isLNat(snd(V1)) -> a__U81(a__isPLNat(V1))
a__isLNat(tail(V1)) -> a__U91(a__isLNat(V1))
a__isLNat(take(V1,V2)) -> a__U101(a__isNatural(V1),V2)
a__isNatural(0()) -> tt()
a__isNatural(head(V1)) -> a__U111(a__isLNat(V1))
a__isNatural(s(V1)) -> a__U121(a__isNatural(V1))
a__isNatural(sel(V1,V2)) -> a__U131(a__isNatural(V1),V2)
a__isPLNat(pair(V1,V2)) -> a__U141(a__isLNat(V1),V2)
a__isPLNat(splitAt(V1,V2)) -> a__U151(a__isNatural(V1),V2)
a__natsFrom(N) -> a__U161(a__isNatural(N),N)
a__sel(N,XS) -> a__U171(a__isNatural(N),N,XS)
a__snd(pair(X,Y)) -> a__U181(a__isLNat(X),Y)
a__splitAt(0(),XS) -> a__U191(a__isLNat(XS),XS)
a__splitAt(s(N),cons(X,XS)) -> a__U201(a__isNatural(N),N,X,XS)
a__tail(cons(N,XS)) -> a__U211(a__isNatural(N),XS)
a__take(N,XS) -> a__U221(a__isNatural(N),N,XS)
mark(U101(X1,X2)) -> a__U101(mark(X1),X2)
mark(U102(X)) -> a__U102(mark(X))
mark(isLNat(X)) -> a__isLNat(X)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(U111(X)) -> a__U111(mark(X))
mark(snd(X)) -> a__snd(mark(X))
mark(splitAt(X1,X2)) -> a__splitAt(mark(X1),mark(X2))
mark(U121(X)) -> a__U121(mark(X))
mark(U131(X1,X2)) -> a__U131(mark(X1),X2)
mark(U132(X)) -> a__U132(mark(X))
mark(U141(X1,X2)) -> a__U141(mark(X1),X2)
mark(U142(X)) -> a__U142(mark(X))
mark(U151(X1,X2)) -> a__U151(mark(X1),X2)
mark(U152(X)) -> a__U152(mark(X))
mark(U161(X1,X2)) -> a__U161(mark(X1),X2)
mark(natsFrom(X)) -> a__natsFrom(mark(X))
mark(U171(X1,X2,X3)) -> a__U171(mark(X1),X2,X3)
mark(U172(X1,X2,X3)) -> a__U172(mark(X1),X2,X3)
mark(head(X)) -> a__head(mark(X))
mark(afterNth(X1,X2)) -> a__afterNth(mark(X1),mark(X2))
mark(U181(X1,X2)) -> a__U181(mark(X1),X2)
mark(U182(X1,X2)) -> a__U182(mark(X1),X2)
mark(U191(X1,X2)) -> a__U191(mark(X1),X2)
mark(U201(X1,X2,X3,X4)) -> a__U201(mark(X1),X2,X3,X4)
mark(U202(X1,X2,X3,X4)) -> a__U202(mark(X1),X2,X3,X4)
mark(isNatural(X)) -> a__isNatural(X)
mark(U203(X1,X2,X3,X4)) -> a__U203(mark(X1),X2,X3,X4)
mark(U204(X1,X2)) -> a__U204(mark(X1),X2)
mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U211(X1,X2)) -> a__U211(mark(X1),X2)
mark(U212(X1,X2)) -> a__U212(mark(X1),X2)
mark(U221(X1,X2,X3)) -> a__U221(mark(X1),X2,X3)
mark(U222(X1,X2,X3)) -> a__U222(mark(X1),X2,X3)
mark(fst(X)) -> a__fst(mark(X))
mark(U31(X1,X2,X3)) -> a__U31(mark(X1),X2,X3)
mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
mark(U42(X)) -> a__U42(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X)) -> a__U52(mark(X))
mark(U61(X)) -> a__U61(mark(X))
mark(U71(X)) -> a__U71(mark(X))
mark(U81(X)) -> a__U81(mark(X))
mark(U91(X)) -> a__U91(mark(X))
mark(isPLNat(X)) -> a__isPLNat(X)
mark(tail(X)) -> a__tail(mark(X))
mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(s(X)) -> s(mark(X))
mark(pair(X1,X2)) -> pair(mark(X1),mark(X2))
mark(nil()) -> nil()
mark(0()) -> 0()
a__U101(X1,X2) -> U101(X1,X2)
a__U102(X) -> U102(X)
a__isLNat(X) -> isLNat(X)
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__U111(X) -> U111(X)
a__snd(X) -> snd(X)
a__splitAt(X1,X2) -> splitAt(X1,X2)
a__U121(X) -> U121(X)
a__U131(X1,X2) -> U131(X1,X2)
a__U132(X) -> U132(X)
a__U141(X1,X2) -> U141(X1,X2)
a__U142(X) -> U142(X)
a__U151(X1,X2) -> U151(X1,X2)
a__U152(X) -> U152(X)
a__U161(X1,X2) -> U161(X1,X2)
a__natsFrom(X) -> natsFrom(X)
a__U171(X1,X2,X3) -> U171(X1,X2,X3)
a__U172(X1,X2,X3) -> U172(X1,X2,X3)
a__head(X) -> head(X)
a__afterNth(X1,X2) -> afterNth(X1,X2)
a__U181(X1,X2) -> U181(X1,X2)
a__U182(X1,X2) -> U182(X1,X2)
a__U191(X1,X2) -> U191(X1,X2)
a__U201(X1,X2,X3,X4) -> U201(X1,X2,X3,X4)
a__U202(X1,X2,X3,X4) -> U202(X1,X2,X3,X4)
a__isNatural(X) -> isNatural(X)
a__U203(X1,X2,X3,X4) -> U203(X1,X2,X3,X4)
a__U204(X1,X2) -> U204(X1,X2)
a__U21(X1,X2,X3) -> U21(X1,X2,X3)
a__U22(X1,X2) -> U22(X1,X2)
a__U211(X1,X2) -> U211(X1,X2)
a__U212(X1,X2) -> U212(X1,X2)
a__U221(X1,X2,X3) -> U221(X1,X2,X3)
a__U222(X1,X2,X3) -> U222(X1,X2,X3)
a__fst(X) -> fst(X)
a__U31(X1,X2,X3) -> U31(X1,X2,X3)
a__U32(X1,X2) -> U32(X1,X2)
a__U41(X1,X2) -> U41(X1,X2)
a__U42(X) -> U42(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X) -> U52(X)
a__U61(X) -> U61(X)
a__U71(X) -> U71(X)
a__U81(X) -> U81(X)
a__U91(X) -> U91(X)
a__isPLNat(X) -> isPLNat(X)
a__tail(X) -> tail(X)
a__take(X1,X2) -> take(X1,X2)
a__sel(X1,X2) -> sel(X1,X2)

Proof:
Open

Tool IRC1

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

stdout:

MAYBE

Tool IRC2

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

stdout:

TIMEOUT

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           TIMEOUT
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  a__U101(tt(), V2) -> a__U102(a__isLNat(V2))
     , a__U102(tt()) -> tt()
     , a__U11(tt(), N, XS) -> a__U12(a__isLNat(XS), N, XS)
     , a__U111(tt()) -> tt()
     , a__U12(tt(), N, XS) -> a__snd(a__splitAt(mark(N), mark(XS)))
     , a__U121(tt()) -> tt()
     , a__U131(tt(), V2) -> a__U132(a__isLNat(V2))
     , a__U132(tt()) -> tt()
     , a__U141(tt(), V2) -> a__U142(a__isLNat(V2))
     , a__U142(tt()) -> tt()
     , a__U151(tt(), V2) -> a__U152(a__isLNat(V2))
     , a__U152(tt()) -> tt()
     , a__U161(tt(), N) -> cons(mark(N), natsFrom(s(N)))
     , a__U171(tt(), N, XS) -> a__U172(a__isLNat(XS), N, XS)
     , a__U172(tt(), N, XS) -> a__head(a__afterNth(mark(N), mark(XS)))
     , a__U181(tt(), Y) -> a__U182(a__isLNat(Y), Y)
     , a__U182(tt(), Y) -> mark(Y)
     , a__U191(tt(), XS) -> pair(nil(), mark(XS))
     , a__U201(tt(), N, X, XS) -> a__U202(a__isNatural(X), N, X, XS)
     , a__U202(tt(), N, X, XS) -> a__U203(a__isLNat(XS), N, X, XS)
     , a__U203(tt(), N, X, XS) ->
       a__U204(a__splitAt(mark(N), mark(XS)), X)
     , a__U204(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS))
     , a__U21(tt(), X, Y) -> a__U22(a__isLNat(Y), X)
     , a__U211(tt(), XS) -> a__U212(a__isLNat(XS), XS)
     , a__U212(tt(), XS) -> mark(XS)
     , a__U22(tt(), X) -> mark(X)
     , a__U221(tt(), N, XS) -> a__U222(a__isLNat(XS), N, XS)
     , a__U222(tt(), N, XS) -> a__fst(a__splitAt(mark(N), mark(XS)))
     , a__U31(tt(), N, XS) -> a__U32(a__isLNat(XS), N)
     , a__U32(tt(), N) -> mark(N)
     , a__U41(tt(), V2) -> a__U42(a__isLNat(V2))
     , a__U42(tt()) -> tt()
     , a__U51(tt(), V2) -> a__U52(a__isLNat(V2))
     , a__U52(tt()) -> tt()
     , a__U61(tt()) -> tt()
     , a__U71(tt()) -> tt()
     , a__U81(tt()) -> tt()
     , a__U91(tt()) -> tt()
     , a__afterNth(N, XS) -> a__U11(a__isNatural(N), N, XS)
     , a__fst(pair(X, Y)) -> a__U21(a__isLNat(X), X, Y)
     , a__head(cons(N, XS)) -> a__U31(a__isNatural(N), N, XS)
     , a__isLNat(nil()) -> tt()
     , a__isLNat(afterNth(V1, V2)) -> a__U41(a__isNatural(V1), V2)
     , a__isLNat(cons(V1, V2)) -> a__U51(a__isNatural(V1), V2)
     , a__isLNat(fst(V1)) -> a__U61(a__isPLNat(V1))
     , a__isLNat(natsFrom(V1)) -> a__U71(a__isNatural(V1))
     , a__isLNat(snd(V1)) -> a__U81(a__isPLNat(V1))
     , a__isLNat(tail(V1)) -> a__U91(a__isLNat(V1))
     , a__isLNat(take(V1, V2)) -> a__U101(a__isNatural(V1), V2)
     , a__isNatural(0()) -> tt()
     , a__isNatural(head(V1)) -> a__U111(a__isLNat(V1))
     , a__isNatural(s(V1)) -> a__U121(a__isNatural(V1))
     , a__isNatural(sel(V1, V2)) -> a__U131(a__isNatural(V1), V2)
     , a__isPLNat(pair(V1, V2)) -> a__U141(a__isLNat(V1), V2)
     , a__isPLNat(splitAt(V1, V2)) -> a__U151(a__isNatural(V1), V2)
     , a__natsFrom(N) -> a__U161(a__isNatural(N), N)
     , a__sel(N, XS) -> a__U171(a__isNatural(N), N, XS)
     , a__snd(pair(X, Y)) -> a__U181(a__isLNat(X), Y)
     , a__splitAt(0(), XS) -> a__U191(a__isLNat(XS), XS)
     , a__splitAt(s(N), cons(X, XS)) ->
       a__U201(a__isNatural(N), N, X, XS)
     , a__tail(cons(N, XS)) -> a__U211(a__isNatural(N), XS)
     , a__take(N, XS) -> a__U221(a__isNatural(N), N, XS)
     , mark(U101(X1, X2)) -> a__U101(mark(X1), X2)
     , mark(U102(X)) -> a__U102(mark(X))
     , mark(isLNat(X)) -> a__isLNat(X)
     , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
     , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
     , mark(U111(X)) -> a__U111(mark(X))
     , mark(snd(X)) -> a__snd(mark(X))
     , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2))
     , mark(U121(X)) -> a__U121(mark(X))
     , mark(U131(X1, X2)) -> a__U131(mark(X1), X2)
     , mark(U132(X)) -> a__U132(mark(X))
     , mark(U141(X1, X2)) -> a__U141(mark(X1), X2)
     , mark(U142(X)) -> a__U142(mark(X))
     , mark(U151(X1, X2)) -> a__U151(mark(X1), X2)
     , mark(U152(X)) -> a__U152(mark(X))
     , mark(U161(X1, X2)) -> a__U161(mark(X1), X2)
     , mark(natsFrom(X)) -> a__natsFrom(mark(X))
     , mark(U171(X1, X2, X3)) -> a__U171(mark(X1), X2, X3)
     , mark(U172(X1, X2, X3)) -> a__U172(mark(X1), X2, X3)
     , mark(head(X)) -> a__head(mark(X))
     , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2))
     , mark(U181(X1, X2)) -> a__U181(mark(X1), X2)
     , mark(U182(X1, X2)) -> a__U182(mark(X1), X2)
     , mark(U191(X1, X2)) -> a__U191(mark(X1), X2)
     , mark(U201(X1, X2, X3, X4)) -> a__U201(mark(X1), X2, X3, X4)
     , mark(U202(X1, X2, X3, X4)) -> a__U202(mark(X1), X2, X3, X4)
     , mark(isNatural(X)) -> a__isNatural(X)
     , mark(U203(X1, X2, X3, X4)) -> a__U203(mark(X1), X2, X3, X4)
     , mark(U204(X1, X2)) -> a__U204(mark(X1), X2)
     , mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3)
     , mark(U22(X1, X2)) -> a__U22(mark(X1), X2)
     , mark(U211(X1, X2)) -> a__U211(mark(X1), X2)
     , mark(U212(X1, X2)) -> a__U212(mark(X1), X2)
     , mark(U221(X1, X2, X3)) -> a__U221(mark(X1), X2, X3)
     , mark(U222(X1, X2, X3)) -> a__U222(mark(X1), X2, X3)
     , mark(fst(X)) -> a__fst(mark(X))
     , mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3)
     , mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
     , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
     , mark(U42(X)) -> a__U42(mark(X))
     , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
     , mark(U52(X)) -> a__U52(mark(X))
     , mark(U61(X)) -> a__U61(mark(X))
     , mark(U71(X)) -> a__U71(mark(X))
     , mark(U81(X)) -> a__U81(mark(X))
     , mark(U91(X)) -> a__U91(mark(X))
     , mark(isPLNat(X)) -> a__isPLNat(X)
     , mark(tail(X)) -> a__tail(mark(X))
     , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
     , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
     , mark(tt()) -> tt()
     , mark(cons(X1, X2)) -> cons(mark(X1), X2)
     , mark(s(X)) -> s(mark(X))
     , mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
     , mark(nil()) -> nil()
     , mark(0()) -> 0()
     , a__U101(X1, X2) -> U101(X1, X2)
     , a__U102(X) -> U102(X)
     , a__isLNat(X) -> isLNat(X)
     , a__U11(X1, X2, X3) -> U11(X1, X2, X3)
     , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
     , a__U111(X) -> U111(X)
     , a__snd(X) -> snd(X)
     , a__splitAt(X1, X2) -> splitAt(X1, X2)
     , a__U121(X) -> U121(X)
     , a__U131(X1, X2) -> U131(X1, X2)
     , a__U132(X) -> U132(X)
     , a__U141(X1, X2) -> U141(X1, X2)
     , a__U142(X) -> U142(X)
     , a__U151(X1, X2) -> U151(X1, X2)
     , a__U152(X) -> U152(X)
     , a__U161(X1, X2) -> U161(X1, X2)
     , a__natsFrom(X) -> natsFrom(X)
     , a__U171(X1, X2, X3) -> U171(X1, X2, X3)
     , a__U172(X1, X2, X3) -> U172(X1, X2, X3)
     , a__head(X) -> head(X)
     , a__afterNth(X1, X2) -> afterNth(X1, X2)
     , a__U181(X1, X2) -> U181(X1, X2)
     , a__U182(X1, X2) -> U182(X1, X2)
     , a__U191(X1, X2) -> U191(X1, X2)
     , a__U201(X1, X2, X3, X4) -> U201(X1, X2, X3, X4)
     , a__U202(X1, X2, X3, X4) -> U202(X1, X2, X3, X4)
     , a__isNatural(X) -> isNatural(X)
     , a__U203(X1, X2, X3, X4) -> U203(X1, X2, X3, X4)
     , a__U204(X1, X2) -> U204(X1, X2)
     , a__U21(X1, X2, X3) -> U21(X1, X2, X3)
     , a__U22(X1, X2) -> U22(X1, X2)
     , a__U211(X1, X2) -> U211(X1, X2)
     , a__U212(X1, X2) -> U212(X1, X2)
     , a__U221(X1, X2, X3) -> U221(X1, X2, X3)
     , a__U222(X1, X2, X3) -> U222(X1, X2, X3)
     , a__fst(X) -> fst(X)
     , a__U31(X1, X2, X3) -> U31(X1, X2, X3)
     , a__U32(X1, X2) -> U32(X1, X2)
     , a__U41(X1, X2) -> U41(X1, X2)
     , a__U42(X) -> U42(X)
     , a__U51(X1, X2) -> U51(X1, X2)
     , a__U52(X) -> U52(X)
     , a__U61(X) -> U61(X)
     , a__U71(X) -> U71(X)
     , a__U81(X) -> U81(X)
     , a__U91(X) -> U91(X)
     , a__isPLNat(X) -> isPLNat(X)
     , a__tail(X) -> tail(X)
     , a__take(X1, X2) -> take(X1, X2)
     , a__sel(X1, X2) -> sel(X1, X2)}

Proof Output:
  Computation stopped due to timeout after 60.0 seconds

Tool RC1

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

stdout:

MAYBE

Tool RC2

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

stdout:

TIMEOUT

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           TIMEOUT
Input Problem:    runtime-complexity with respect to
  Rules:
    {  a__U101(tt(), V2) -> a__U102(a__isLNat(V2))
     , a__U102(tt()) -> tt()
     , a__U11(tt(), N, XS) -> a__U12(a__isLNat(XS), N, XS)
     , a__U111(tt()) -> tt()
     , a__U12(tt(), N, XS) -> a__snd(a__splitAt(mark(N), mark(XS)))
     , a__U121(tt()) -> tt()
     , a__U131(tt(), V2) -> a__U132(a__isLNat(V2))
     , a__U132(tt()) -> tt()
     , a__U141(tt(), V2) -> a__U142(a__isLNat(V2))
     , a__U142(tt()) -> tt()
     , a__U151(tt(), V2) -> a__U152(a__isLNat(V2))
     , a__U152(tt()) -> tt()
     , a__U161(tt(), N) -> cons(mark(N), natsFrom(s(N)))
     , a__U171(tt(), N, XS) -> a__U172(a__isLNat(XS), N, XS)
     , a__U172(tt(), N, XS) -> a__head(a__afterNth(mark(N), mark(XS)))
     , a__U181(tt(), Y) -> a__U182(a__isLNat(Y), Y)
     , a__U182(tt(), Y) -> mark(Y)
     , a__U191(tt(), XS) -> pair(nil(), mark(XS))
     , a__U201(tt(), N, X, XS) -> a__U202(a__isNatural(X), N, X, XS)
     , a__U202(tt(), N, X, XS) -> a__U203(a__isLNat(XS), N, X, XS)
     , a__U203(tt(), N, X, XS) ->
       a__U204(a__splitAt(mark(N), mark(XS)), X)
     , a__U204(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS))
     , a__U21(tt(), X, Y) -> a__U22(a__isLNat(Y), X)
     , a__U211(tt(), XS) -> a__U212(a__isLNat(XS), XS)
     , a__U212(tt(), XS) -> mark(XS)
     , a__U22(tt(), X) -> mark(X)
     , a__U221(tt(), N, XS) -> a__U222(a__isLNat(XS), N, XS)
     , a__U222(tt(), N, XS) -> a__fst(a__splitAt(mark(N), mark(XS)))
     , a__U31(tt(), N, XS) -> a__U32(a__isLNat(XS), N)
     , a__U32(tt(), N) -> mark(N)
     , a__U41(tt(), V2) -> a__U42(a__isLNat(V2))
     , a__U42(tt()) -> tt()
     , a__U51(tt(), V2) -> a__U52(a__isLNat(V2))
     , a__U52(tt()) -> tt()
     , a__U61(tt()) -> tt()
     , a__U71(tt()) -> tt()
     , a__U81(tt()) -> tt()
     , a__U91(tt()) -> tt()
     , a__afterNth(N, XS) -> a__U11(a__isNatural(N), N, XS)
     , a__fst(pair(X, Y)) -> a__U21(a__isLNat(X), X, Y)
     , a__head(cons(N, XS)) -> a__U31(a__isNatural(N), N, XS)
     , a__isLNat(nil()) -> tt()
     , a__isLNat(afterNth(V1, V2)) -> a__U41(a__isNatural(V1), V2)
     , a__isLNat(cons(V1, V2)) -> a__U51(a__isNatural(V1), V2)
     , a__isLNat(fst(V1)) -> a__U61(a__isPLNat(V1))
     , a__isLNat(natsFrom(V1)) -> a__U71(a__isNatural(V1))
     , a__isLNat(snd(V1)) -> a__U81(a__isPLNat(V1))
     , a__isLNat(tail(V1)) -> a__U91(a__isLNat(V1))
     , a__isLNat(take(V1, V2)) -> a__U101(a__isNatural(V1), V2)
     , a__isNatural(0()) -> tt()
     , a__isNatural(head(V1)) -> a__U111(a__isLNat(V1))
     , a__isNatural(s(V1)) -> a__U121(a__isNatural(V1))
     , a__isNatural(sel(V1, V2)) -> a__U131(a__isNatural(V1), V2)
     , a__isPLNat(pair(V1, V2)) -> a__U141(a__isLNat(V1), V2)
     , a__isPLNat(splitAt(V1, V2)) -> a__U151(a__isNatural(V1), V2)
     , a__natsFrom(N) -> a__U161(a__isNatural(N), N)
     , a__sel(N, XS) -> a__U171(a__isNatural(N), N, XS)
     , a__snd(pair(X, Y)) -> a__U181(a__isLNat(X), Y)
     , a__splitAt(0(), XS) -> a__U191(a__isLNat(XS), XS)
     , a__splitAt(s(N), cons(X, XS)) ->
       a__U201(a__isNatural(N), N, X, XS)
     , a__tail(cons(N, XS)) -> a__U211(a__isNatural(N), XS)
     , a__take(N, XS) -> a__U221(a__isNatural(N), N, XS)
     , mark(U101(X1, X2)) -> a__U101(mark(X1), X2)
     , mark(U102(X)) -> a__U102(mark(X))
     , mark(isLNat(X)) -> a__isLNat(X)
     , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
     , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
     , mark(U111(X)) -> a__U111(mark(X))
     , mark(snd(X)) -> a__snd(mark(X))
     , mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2))
     , mark(U121(X)) -> a__U121(mark(X))
     , mark(U131(X1, X2)) -> a__U131(mark(X1), X2)
     , mark(U132(X)) -> a__U132(mark(X))
     , mark(U141(X1, X2)) -> a__U141(mark(X1), X2)
     , mark(U142(X)) -> a__U142(mark(X))
     , mark(U151(X1, X2)) -> a__U151(mark(X1), X2)
     , mark(U152(X)) -> a__U152(mark(X))
     , mark(U161(X1, X2)) -> a__U161(mark(X1), X2)
     , mark(natsFrom(X)) -> a__natsFrom(mark(X))
     , mark(U171(X1, X2, X3)) -> a__U171(mark(X1), X2, X3)
     , mark(U172(X1, X2, X3)) -> a__U172(mark(X1), X2, X3)
     , mark(head(X)) -> a__head(mark(X))
     , mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2))
     , mark(U181(X1, X2)) -> a__U181(mark(X1), X2)
     , mark(U182(X1, X2)) -> a__U182(mark(X1), X2)
     , mark(U191(X1, X2)) -> a__U191(mark(X1), X2)
     , mark(U201(X1, X2, X3, X4)) -> a__U201(mark(X1), X2, X3, X4)
     , mark(U202(X1, X2, X3, X4)) -> a__U202(mark(X1), X2, X3, X4)
     , mark(isNatural(X)) -> a__isNatural(X)
     , mark(U203(X1, X2, X3, X4)) -> a__U203(mark(X1), X2, X3, X4)
     , mark(U204(X1, X2)) -> a__U204(mark(X1), X2)
     , mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3)
     , mark(U22(X1, X2)) -> a__U22(mark(X1), X2)
     , mark(U211(X1, X2)) -> a__U211(mark(X1), X2)
     , mark(U212(X1, X2)) -> a__U212(mark(X1), X2)
     , mark(U221(X1, X2, X3)) -> a__U221(mark(X1), X2, X3)
     , mark(U222(X1, X2, X3)) -> a__U222(mark(X1), X2, X3)
     , mark(fst(X)) -> a__fst(mark(X))
     , mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3)
     , mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
     , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
     , mark(U42(X)) -> a__U42(mark(X))
     , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
     , mark(U52(X)) -> a__U52(mark(X))
     , mark(U61(X)) -> a__U61(mark(X))
     , mark(U71(X)) -> a__U71(mark(X))
     , mark(U81(X)) -> a__U81(mark(X))
     , mark(U91(X)) -> a__U91(mark(X))
     , mark(isPLNat(X)) -> a__isPLNat(X)
     , mark(tail(X)) -> a__tail(mark(X))
     , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
     , mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
     , mark(tt()) -> tt()
     , mark(cons(X1, X2)) -> cons(mark(X1), X2)
     , mark(s(X)) -> s(mark(X))
     , mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
     , mark(nil()) -> nil()
     , mark(0()) -> 0()
     , a__U101(X1, X2) -> U101(X1, X2)
     , a__U102(X) -> U102(X)
     , a__isLNat(X) -> isLNat(X)
     , a__U11(X1, X2, X3) -> U11(X1, X2, X3)
     , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
     , a__U111(X) -> U111(X)
     , a__snd(X) -> snd(X)
     , a__splitAt(X1, X2) -> splitAt(X1, X2)
     , a__U121(X) -> U121(X)
     , a__U131(X1, X2) -> U131(X1, X2)
     , a__U132(X) -> U132(X)
     , a__U141(X1, X2) -> U141(X1, X2)
     , a__U142(X) -> U142(X)
     , a__U151(X1, X2) -> U151(X1, X2)
     , a__U152(X) -> U152(X)
     , a__U161(X1, X2) -> U161(X1, X2)
     , a__natsFrom(X) -> natsFrom(X)
     , a__U171(X1, X2, X3) -> U171(X1, X2, X3)
     , a__U172(X1, X2, X3) -> U172(X1, X2, X3)
     , a__head(X) -> head(X)
     , a__afterNth(X1, X2) -> afterNth(X1, X2)
     , a__U181(X1, X2) -> U181(X1, X2)
     , a__U182(X1, X2) -> U182(X1, X2)
     , a__U191(X1, X2) -> U191(X1, X2)
     , a__U201(X1, X2, X3, X4) -> U201(X1, X2, X3, X4)
     , a__U202(X1, X2, X3, X4) -> U202(X1, X2, X3, X4)
     , a__isNatural(X) -> isNatural(X)
     , a__U203(X1, X2, X3, X4) -> U203(X1, X2, X3, X4)
     , a__U204(X1, X2) -> U204(X1, X2)
     , a__U21(X1, X2, X3) -> U21(X1, X2, X3)
     , a__U22(X1, X2) -> U22(X1, X2)
     , a__U211(X1, X2) -> U211(X1, X2)
     , a__U212(X1, X2) -> U212(X1, X2)
     , a__U221(X1, X2, X3) -> U221(X1, X2, X3)
     , a__U222(X1, X2, X3) -> U222(X1, X2, X3)
     , a__fst(X) -> fst(X)
     , a__U31(X1, X2, X3) -> U31(X1, X2, X3)
     , a__U32(X1, X2) -> U32(X1, X2)
     , a__U41(X1, X2) -> U41(X1, X2)
     , a__U42(X) -> U42(X)
     , a__U51(X1, X2) -> U51(X1, X2)
     , a__U52(X) -> U52(X)
     , a__U61(X) -> U61(X)
     , a__U71(X) -> U71(X)
     , a__U81(X) -> U81(X)
     , a__U91(X) -> U91(X)
     , a__isPLNat(X) -> isPLNat(X)
     , a__tail(X) -> tail(X)
     , a__take(X1, X2) -> take(X1, X2)
     , a__sel(X1, X2) -> sel(X1, X2)}

Proof Output:
  Computation stopped due to timeout after 60.0 seconds