(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))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U81(X1, X2, X3, X4)) → a__U81(mark(X1), X2, X3, X4)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatural(X)) → a__isNatural(X)
mark(isLNat(X)) → a__isLNat(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, X3) → U101(X1, X2, X3)
a__fst(X) → fst(X)
a__splitAt(X1, X2) → splitAt(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__snd(X) → snd(X)
a__U21(X1, X2) → U21(X1, X2)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2) → U41(X1, X2)
a__natsFrom(X) → natsFrom(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__head(X) → head(X)
a__afterNth(X1, X2) → afterNth(X1, X2)
a__U61(X1, X2) → U61(X1, X2)
a__U71(X1, X2) → U71(X1, X2)
a__U81(X1, X2, X3, X4) → U81(X1, X2, X3, X4)
a__U82(X1, X2) → U82(X1, X2)
a__U91(X1, X2) → U91(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatural(X) → isNatural(X)
a__isLNat(X) → isLNat(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)
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__U101(tt, N, XS) → A__FST(a__splitAt(mark(N), mark(XS)))
A__U101(tt, N, XS) → A__SPLITAT(mark(N), mark(XS))
A__U101(tt, N, XS) → MARK(N)
A__U101(tt, N, XS) → MARK(XS)
A__U11(tt, N, XS) → A__SND(a__splitAt(mark(N), mark(XS)))
A__U11(tt, N, XS) → A__SPLITAT(mark(N), mark(XS))
A__U11(tt, N, XS) → MARK(N)
A__U11(tt, N, XS) → MARK(XS)
A__U21(tt, X) → MARK(X)
A__U31(tt, N) → MARK(N)
A__U41(tt, N) → MARK(N)
A__U51(tt, N, XS) → A__HEAD(a__afterNth(mark(N), mark(XS)))
A__U51(tt, N, XS) → A__AFTERNTH(mark(N), mark(XS))
A__U51(tt, N, XS) → MARK(N)
A__U51(tt, N, XS) → MARK(XS)
A__U61(tt, Y) → MARK(Y)
A__U71(tt, XS) → MARK(XS)
A__U81(tt, N, X, XS) → A__U82(a__splitAt(mark(N), mark(XS)), X)
A__U81(tt, N, X, XS) → A__SPLITAT(mark(N), mark(XS))
A__U81(tt, N, X, XS) → MARK(N)
A__U81(tt, N, X, XS) → MARK(XS)
A__U82(pair(YS, ZS), X) → MARK(X)
A__U82(pair(YS, ZS), X) → MARK(ZS)
A__U91(tt, XS) → MARK(XS)
A__AFTERNTH(N, XS) → A__U11(a__and(a__isNatural(N), isLNat(XS)), N, XS)
A__AFTERNTH(N, XS) → A__AND(a__isNatural(N), isLNat(XS))
A__AFTERNTH(N, XS) → A__ISNATURAL(N)
A__AND(tt, X) → MARK(X)
A__FST(pair(X, Y)) → A__U21(a__and(a__isLNat(X), isLNat(Y)), X)
A__FST(pair(X, Y)) → A__AND(a__isLNat(X), isLNat(Y))
A__FST(pair(X, Y)) → A__ISLNAT(X)
A__HEAD(cons(N, XS)) → A__U31(a__and(a__isNatural(N), isLNat(XS)), N)
A__HEAD(cons(N, XS)) → A__AND(a__isNatural(N), isLNat(XS))
A__HEAD(cons(N, XS)) → A__ISNATURAL(N)
A__ISLNAT(afterNth(V1, V2)) → A__AND(a__isNatural(V1), isLNat(V2))
A__ISLNAT(afterNth(V1, V2)) → A__ISNATURAL(V1)
A__ISLNAT(cons(V1, V2)) → A__AND(a__isNatural(V1), isLNat(V2))
A__ISLNAT(cons(V1, V2)) → A__ISNATURAL(V1)
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__ISLNAT(take(V1, V2)) → A__ISNATURAL(V1)
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__ISNATURAL(sel(V1, V2)) → A__ISNATURAL(V1)
A__ISPLNAT(pair(V1, V2)) → A__AND(a__isLNat(V1), isLNat(V2))
A__ISPLNAT(pair(V1, V2)) → A__ISLNAT(V1)
A__ISPLNAT(splitAt(V1, V2)) → A__AND(a__isNatural(V1), isLNat(V2))
A__ISPLNAT(splitAt(V1, V2)) → A__ISNATURAL(V1)
A__NATSFROM(N) → A__U41(a__isNatural(N), N)
A__NATSFROM(N) → A__ISNATURAL(N)
A__SEL(N, XS) → A__U51(a__and(a__isNatural(N), isLNat(XS)), N, XS)
A__SEL(N, XS) → A__AND(a__isNatural(N), isLNat(XS))
A__SEL(N, XS) → A__ISNATURAL(N)
A__SND(pair(X, Y)) → A__U61(a__and(a__isLNat(X), isLNat(Y)), Y)
A__SND(pair(X, Y)) → A__AND(a__isLNat(X), isLNat(Y))
A__SND(pair(X, Y)) → A__ISLNAT(X)
A__SPLITAT(0, XS) → A__U71(a__isLNat(XS), XS)
A__SPLITAT(0, XS) → A__ISLNAT(XS)
A__SPLITAT(s(N), cons(X, XS)) → A__U81(a__and(a__isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)
A__SPLITAT(s(N), cons(X, XS)) → A__AND(a__isNatural(N), and(isNatural(X), isLNat(XS)))
A__SPLITAT(s(N), cons(X, XS)) → A__ISNATURAL(N)
A__TAIL(cons(N, XS)) → A__U91(a__and(a__isNatural(N), isLNat(XS)), XS)
A__TAIL(cons(N, XS)) → A__AND(a__isNatural(N), isLNat(XS))
A__TAIL(cons(N, XS)) → A__ISNATURAL(N)
A__TAKE(N, XS) → A__U101(a__and(a__isNatural(N), isLNat(XS)), N, XS)
A__TAKE(N, XS) → A__AND(a__isNatural(N), isLNat(XS))
A__TAKE(N, XS) → A__ISNATURAL(N)
MARK(U101(X1, X2, X3)) → A__U101(mark(X1), X2, X3)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → A__FST(mark(X))
MARK(fst(X)) → MARK(X)
MARK(splitAt(X1, X2)) → A__SPLITAT(mark(X1), mark(X2))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(snd(X)) → A__SND(mark(X))
MARK(snd(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(natsFrom(X)) → A__NATSFROM(mark(X))
MARK(natsFrom(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(head(X)) → A__HEAD(mark(X))
MARK(head(X)) → MARK(X)
MARK(afterNth(X1, X2)) → A__AFTERNTH(mark(X1), mark(X2))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U81(X1, X2, X3, X4)) → A__U81(mark(X1), X2, X3, X4)
MARK(U81(X1, X2, X3, X4)) → MARK(X1)
MARK(U82(X1, X2)) → A__U82(mark(X1), X2)
MARK(U82(X1, X2)) → MARK(X1)
MARK(U91(X1, X2)) → A__U91(mark(X1), X2)
MARK(U91(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatural(X)) → A__ISNATURAL(X)
MARK(isLNat(X)) → A__ISLNAT(X)
MARK(isPLNat(X)) → A__ISPLNAT(X)
MARK(tail(X)) → A__TAIL(mark(X))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
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))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U81(X1, X2, X3, X4)) → a__U81(mark(X1), X2, X3, X4)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatural(X)) → a__isNatural(X)
mark(isLNat(X)) → a__isLNat(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, X3) → U101(X1, X2, X3)
a__fst(X) → fst(X)
a__splitAt(X1, X2) → splitAt(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__snd(X) → snd(X)
a__U21(X1, X2) → U21(X1, X2)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2) → U41(X1, X2)
a__natsFrom(X) → natsFrom(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__head(X) → head(X)
a__afterNth(X1, X2) → afterNth(X1, X2)
a__U61(X1, X2) → U61(X1, X2)
a__U71(X1, X2) → U71(X1, X2)
a__U81(X1, X2, X3, X4) → U81(X1, X2, X3, X4)
a__U82(X1, X2) → U82(X1, X2)
a__U91(X1, X2) → U91(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatural(X) → isNatural(X)
a__isLNat(X) → isLNat(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)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.