(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__U11(tt, N, X, XS) → a__U12(a__splitAt(mark(N), mark(XS)), X)
a__U12(pair(YS, ZS), X) → pair(cons(mark(X), YS), mark(ZS))
a__afterNth(N, XS) → a__snd(a__splitAt(mark(N), mark(XS)))
a__and(tt, X) → mark(X)
a__fst(pair(X, Y)) → mark(X)
a__head(cons(N, XS)) → mark(N)
a__natsFrom(N) → cons(mark(N), natsFrom(s(N)))
a__sel(N, XS) → a__head(a__afterNth(mark(N), mark(XS)))
a__snd(pair(X, Y)) → mark(Y)
a__splitAt(0, XS) → pair(nil, mark(XS))
a__splitAt(s(N), cons(X, XS)) → a__U11(tt, N, X, XS)
a__tail(cons(N, XS)) → mark(XS)
a__take(N, XS) → a__fst(a__splitAt(mark(N), mark(XS)))
mark(U11(X1, X2, X3, X4)) → a__U11(mark(X1), X2, X3, X4)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(splitAt(X1, X2)) → a__splitAt(mark(X1), mark(X2))
mark(afterNth(X1, X2)) → a__afterNth(mark(X1), mark(X2))
mark(snd(X)) → a__snd(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(fst(X)) → a__fst(mark(X))
mark(head(X)) → a__head(mark(X))
mark(natsFrom(X)) → a__natsFrom(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(tt) → tt
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__U11(X1, X2, X3, X4) → U11(X1, X2, X3, X4)
a__U12(X1, X2) → U12(X1, X2)
a__splitAt(X1, X2) → splitAt(X1, X2)
a__afterNth(X1, X2) → afterNth(X1, X2)
a__snd(X) → snd(X)
a__and(X1, X2) → and(X1, X2)
a__fst(X) → fst(X)
a__head(X) → head(X)
a__natsFrom(X) → natsFrom(X)
a__sel(X1, X2) → sel(X1, X2)
a__tail(X) → tail(X)
a__take(X1, X2) → take(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__U11(tt, N, X, XS) → A__U12(a__splitAt(mark(N), mark(XS)), X)
A__U11(tt, N, X, XS) → A__SPLITAT(mark(N), mark(XS))
A__U11(tt, N, X, XS) → MARK(N)
A__U11(tt, N, X, XS) → MARK(XS)
A__U12(pair(YS, ZS), X) → MARK(X)
A__U12(pair(YS, ZS), X) → MARK(ZS)
A__AFTERNTH(N, XS) → A__SND(a__splitAt(mark(N), mark(XS)))
A__AFTERNTH(N, XS) → A__SPLITAT(mark(N), mark(XS))
A__AFTERNTH(N, XS) → MARK(N)
A__AFTERNTH(N, XS) → MARK(XS)
A__AND(tt, X) → MARK(X)
A__FST(pair(X, Y)) → MARK(X)
A__HEAD(cons(N, XS)) → MARK(N)
A__NATSFROM(N) → MARK(N)
A__SEL(N, XS) → A__HEAD(a__afterNth(mark(N), mark(XS)))
A__SEL(N, XS) → A__AFTERNTH(mark(N), mark(XS))
A__SEL(N, XS) → MARK(N)
A__SEL(N, XS) → MARK(XS)
A__SND(pair(X, Y)) → MARK(Y)
A__SPLITAT(0, XS) → MARK(XS)
A__SPLITAT(s(N), cons(X, XS)) → A__U11(tt, N, X, XS)
A__TAIL(cons(N, XS)) → MARK(XS)
A__TAKE(N, XS) → A__FST(a__splitAt(mark(N), mark(XS)))
A__TAKE(N, XS) → A__SPLITAT(mark(N), mark(XS))
A__TAKE(N, XS) → MARK(N)
A__TAKE(N, XS) → MARK(XS)
MARK(U11(X1, X2, X3, X4)) → A__U11(mark(X1), X2, X3, X4)
MARK(U11(X1, X2, X3, X4)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → A__SPLITAT(mark(X1), mark(X2))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(afterNth(X1, X2)) → A__AFTERNTH(mark(X1), mark(X2))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(snd(X)) → A__SND(mark(X))
MARK(snd(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(fst(X)) → A__FST(mark(X))
MARK(fst(X)) → MARK(X)
MARK(head(X)) → A__HEAD(mark(X))
MARK(head(X)) → MARK(X)
MARK(natsFrom(X)) → A__NATSFROM(mark(X))
MARK(natsFrom(X)) → MARK(X)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
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(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__U11(tt, N, X, XS) → a__U12(a__splitAt(mark(N), mark(XS)), X)
a__U12(pair(YS, ZS), X) → pair(cons(mark(X), YS), mark(ZS))
a__afterNth(N, XS) → a__snd(a__splitAt(mark(N), mark(XS)))
a__and(tt, X) → mark(X)
a__fst(pair(X, Y)) → mark(X)
a__head(cons(N, XS)) → mark(N)
a__natsFrom(N) → cons(mark(N), natsFrom(s(N)))
a__sel(N, XS) → a__head(a__afterNth(mark(N), mark(XS)))
a__snd(pair(X, Y)) → mark(Y)
a__splitAt(0, XS) → pair(nil, mark(XS))
a__splitAt(s(N), cons(X, XS)) → a__U11(tt, N, X, XS)
a__tail(cons(N, XS)) → mark(XS)
a__take(N, XS) → a__fst(a__splitAt(mark(N), mark(XS)))
mark(U11(X1, X2, X3, X4)) → a__U11(mark(X1), X2, X3, X4)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(splitAt(X1, X2)) → a__splitAt(mark(X1), mark(X2))
mark(afterNth(X1, X2)) → a__afterNth(mark(X1), mark(X2))
mark(snd(X)) → a__snd(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(fst(X)) → a__fst(mark(X))
mark(head(X)) → a__head(mark(X))
mark(natsFrom(X)) → a__natsFrom(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(tt) → tt
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__U11(X1, X2, X3, X4) → U11(X1, X2, X3, X4)
a__U12(X1, X2) → U12(X1, X2)
a__splitAt(X1, X2) → splitAt(X1, X2)
a__afterNth(X1, X2) → afterNth(X1, X2)
a__snd(X) → snd(X)
a__and(X1, X2) → and(X1, X2)
a__fst(X) → fst(X)
a__head(X) → head(X)
a__natsFrom(X) → natsFrom(X)
a__sel(X1, X2) → sel(X1, X2)
a__tail(X) → tail(X)
a__take(X1, X2) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.