(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__U101(tt, N, XS) → a__fst(a__splitAt(mark(N), mark(XS)))
a__U11(tt, N, XS) → a__snd(a__splitAt(mark(N), mark(XS)))
a__U21(tt, X) → mark(X)
a__U31(tt, N) → mark(N)
a__U41(tt, N) → cons(mark(N), natsFrom(s(N)))
a__U51(tt, N, XS) → a__head(a__afterNth(mark(N), mark(XS)))
a__U61(tt, Y) → mark(Y)
a__U71(tt, XS) → pair(nil, mark(XS))
a__U81(tt, N, X, XS) → a__U82(a__splitAt(mark(N), mark(XS)), X)
a__U82(pair(YS, ZS), X) → pair(cons(mark(X), YS), mark(ZS))
a__U91(tt, XS) → mark(XS)
a__afterNth(N, XS) → a__U11(a__and(a__isNatural(N), isLNat(XS)), N, XS)
a__and(tt, X) → mark(X)
a__fst(pair(X, Y)) → a__U21(a__and(a__isLNat(X), isLNat(Y)), X)
a__head(cons(N, XS)) → a__U31(a__and(a__isNatural(N), isLNat(XS)), N)
a__isLNat(nil) → tt
a__isLNat(afterNth(V1, V2)) → a__and(a__isNatural(V1), isLNat(V2))
a__isLNat(cons(V1, V2)) → a__and(a__isNatural(V1), isLNat(V2))
a__isLNat(fst(V1)) → a__isPLNat(V1)
a__isLNat(natsFrom(V1)) → a__isNatural(V1)
a__isLNat(snd(V1)) → a__isPLNat(V1)
a__isLNat(tail(V1)) → a__isLNat(V1)
a__isLNat(take(V1, V2)) → a__and(a__isNatural(V1), isLNat(V2))
a__isNatural(0) → tt
a__isNatural(head(V1)) → a__isLNat(V1)
a__isNatural(s(V1)) → a__isNatural(V1)
a__isNatural(sel(V1, V2)) → a__and(a__isNatural(V1), isLNat(V2))
a__isPLNat(pair(V1, V2)) → a__and(a__isLNat(V1), isLNat(V2))
a__isPLNat(splitAt(V1, V2)) → a__and(a__isNatural(V1), isLNat(V2))
a__natsFrom(N) → a__U41(a__isNatural(N), N)
a__sel(N, XS) → a__U51(a__and(a__isNatural(N), isLNat(XS)), N, XS)
a__snd(pair(X, Y)) → a__U61(a__and(a__isLNat(X), isLNat(Y)), Y)
a__splitAt(0, XS) → a__U71(a__isLNat(XS), XS)
a__splitAt(s(N), cons(X, XS)) → a__U81(a__and(a__isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)
a__tail(cons(N, XS)) → a__U91(a__and(a__isNatural(N), isLNat(XS)), XS)
a__take(N, XS) → a__U101(a__and(a__isNatural(N), isLNat(XS)), N, XS)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(fst(X)) → a__fst(mark(X))
mark(splitAt(X1, X2)) → a__splitAt(mark(X1), mark(X2))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(snd(X)) → a__snd(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(natsFrom(X)) → a__natsFrom(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(head(X)) → a__head(mark(X))
mark(afterNth(X1, X2)) → a__afterNth(mark(X1), mark(X2))