Termination w.r.t. Q of the given QTRS could not be shown:

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(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:

ACTIVE(U11(tt, N, X, XS)) → MARK(U12(splitAt(N, XS), X))
ACTIVE(U11(tt, N, X, XS)) → U12¹(splitAt(N, XS), X)
ACTIVE(U11(tt, N, X, XS)) → SPLITAT(N, XS)
ACTIVE(U12(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
ACTIVE(U12(pair(YS, ZS), X)) → PAIR(cons(X, YS), ZS)
ACTIVE(U12(pair(YS, ZS), X)) → CONS(X, YS)
ACTIVE(afterNth(N, XS)) → MARK(snd(splitAt(N, XS)))
ACTIVE(afterNth(N, XS)) → SND(splitAt(N, XS))
ACTIVE(afterNth(N, XS)) → SPLITAT(N, XS)
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(fst(pair(X, Y))) → MARK(X)
ACTIVE(head(cons(N, XS))) → MARK(N)
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
ACTIVE(natsFrom(N)) → CONS(N, natsFrom(s(N)))
ACTIVE(natsFrom(N)) → NATSFROM(s(N))
ACTIVE(natsFrom(N)) → S(N)
ACTIVE(sel(N, XS)) → MARK(head(afterNth(N, XS)))
ACTIVE(sel(N, XS)) → HEAD(afterNth(N, XS))
ACTIVE(sel(N, XS)) → AFTERNTH(N, XS)
ACTIVE(snd(pair(X, Y))) → MARK(Y)
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
ACTIVE(splitAt(0, XS)) → PAIR(nil, XS)
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U11(tt, N, X, XS))
ACTIVE(splitAt(s(N), cons(X, XS))) → U11¹(tt, N, X, XS)
ACTIVE(tail(cons(N, XS))) → MARK(XS)
ACTIVE(take(N, XS)) → MARK(fst(splitAt(N, XS)))
ACTIVE(take(N, XS)) → FST(splitAt(N, XS))
ACTIVE(take(N, XS)) → SPLITAT(N, XS)
MARK(U11(X1, X2, X3, X4)) → ACTIVE(U11(mark(X1), X2, X3, X4))
MARK(U11(X1, X2, X3, X4)) → U11¹(mark(X1), X2, X3, X4)
MARK(U11(X1, X2, X3, X4)) → MARK(X1)
MARK(tt) → ACTIVE(tt)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(U12(X1, X2)) → U12¹(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
MARK(splitAt(X1, X2)) → SPLITAT(mark(X1), mark(X2))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(pair(X1, X2)) → ACTIVE(pair(mark(X1), mark(X2)))
MARK(pair(X1, X2)) → PAIR(mark(X1), mark(X2))
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
MARK(afterNth(X1, X2)) → AFTERNTH(mark(X1), mark(X2))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(snd(X)) → SND(mark(X))
MARK(snd(X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(and(X1, X2)) → AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(fst(X)) → FST(mark(X))
MARK(fst(X)) → MARK(X)
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(head(X)) → HEAD(mark(X))
MARK(head(X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
MARK(natsFrom(X)) → NATSFROM(mark(X))
MARK(natsFrom(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → SEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(0) → ACTIVE(0)
MARK(nil) → ACTIVE(nil)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(tail(X)) → TAIL(mark(X))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
U11¹(mark(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, mark(X2), X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, X2, mark(X3), X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, X2, X3, mark(X4)) → U11¹(X1, X2, X3, X4)
U11¹(active(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, active(X2), X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, X2, active(X3), X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, X2, X3, active(X4)) → U11¹(X1, X2, X3, X4)
U12¹(mark(X1), X2) → U12¹(X1, X2)
U12¹(X1, mark(X2)) → U12¹(X1, X2)
U12¹(active(X1), X2) → U12¹(X1, X2)
U12¹(X1, active(X2)) → U12¹(X1, X2)
SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)
SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)
PAIR(mark(X1), X2) → PAIR(X1, X2)
PAIR(X1, mark(X2)) → PAIR(X1, X2)
PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)
SND(mark(X)) → SND(X)
SND(active(X)) → SND(X)
AND(mark(X1), X2) → AND(X1, X2)
AND(X1, mark(X2)) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)
FST(mark(X)) → FST(X)
FST(active(X)) → FST(X)
HEAD(mark(X)) → HEAD(X)
HEAD(active(X)) → HEAD(X)
NATSFROM(mark(X)) → NATSFROM(X)
NATSFROM(active(X)) → NATSFROM(X)
S(mark(X)) → S(X)
S(active(X)) → S(X)
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)
TAIL(mark(X)) → TAIL(X)
TAIL(active(X)) → TAIL(X)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 16 SCCs with 33 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  TAKE(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

mark₁ > TAKE₂

Status:

TAKE₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(X1, active(X2)) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  TAKE(x2)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > TAKE₁

Status:

TAKE₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAKE(active(X1), X2) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(active(X1), X2) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[TAKE₂, active₁]

Status:

TAKE₂: [2,1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(12) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(13) TRUE

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAIL(active(X)) → TAIL(X)
TAIL(mark(X)) → TAIL(X)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(active(X)) → TAIL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAIL(x1)  =  TAIL(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[TAIL₁, active₁]

Status:

TAIL₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAIL(mark(X)) → TAIL(X)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(mark(X)) → TAIL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAIL(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(19) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(20) TRUE

(21) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(mark(X1), X2) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL(x1, x2)  =  SEL(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

mark₁ > SEL₂

Status:

SEL₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(23) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(X1, active(X2)) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL(x1, x2)  =  SEL(x2)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > SEL₁

Status:

SEL₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SEL(active(X1), X2) → SEL(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(active(X1), X2) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[SEL₂, active₁]

Status:

SEL₂: [2,1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(28) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(29) TRUE

(30) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[S₁, active₁]

Status:

S₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(34) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(35) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(36) TRUE

(37) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NATSFROM(active(X)) → NATSFROM(X)
NATSFROM(mark(X)) → NATSFROM(X)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NATSFROM(active(X)) → NATSFROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
NATSFROM(x1)  =  NATSFROM(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[NATSFROM₁, active₁]

Status:

NATSFROM₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NATSFROM(mark(X)) → NATSFROM(X)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NATSFROM(mark(X)) → NATSFROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
NATSFROM(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(42) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(43) TRUE

(44) Obligation:

Q DP problem:
The TRS P consists of the following rules:

HEAD(active(X)) → HEAD(X)
HEAD(mark(X)) → HEAD(X)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(active(X)) → HEAD(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
HEAD(x1)  =  HEAD(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[HEAD₁, active₁]

Status:

HEAD₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

Q DP problem:
The TRS P consists of the following rules:

HEAD(mark(X)) → HEAD(X)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(mark(X)) → HEAD(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
HEAD(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(48) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(49) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(50) TRUE

(51) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FST(active(X)) → FST(X)
FST(mark(X)) → FST(X)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FST(active(X)) → FST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FST(x1)  =  FST(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[FST₁, active₁]

Status:

FST₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(53) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FST(mark(X)) → FST(X)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FST(mark(X)) → FST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FST(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(55) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(56) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(57) TRUE

(58) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AND(X1, mark(X2)) → AND(X1, X2)
AND(mark(X1), X2) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(X1, mark(X2)) → AND(X1, X2)
AND(mark(X1), X2) → AND(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2)  =  AND(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

mark₁ > AND₂

Status:

AND₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(60) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(X1, active(X2)) → AND(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2)  =  AND(x2)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > AND₁

Status:

AND₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(62) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AND(active(X1), X2) → AND(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(active(X1), X2) → AND(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[AND₂, active₁]

Status:

AND₂: [2,1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(64) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(65) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(66) TRUE

(67) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SND(active(X)) → SND(X)
SND(mark(X)) → SND(X)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(68) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SND(active(X)) → SND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SND(x1)  =  SND(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[SND₁, active₁]

Status:

SND₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(69) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SND(mark(X)) → SND(X)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(70) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SND(mark(X)) → SND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SND(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(71) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(72) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(73) TRUE

(74) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(75) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AFTERNTH(x1, x2)  =  AFTERNTH(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

mark₁ > AFTERNTH₂

Status:

AFTERNTH₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(76) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(77) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AFTERNTH(x1, x2)  =  AFTERNTH(x2)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > AFTERNTH₁

Status:

AFTERNTH₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(78) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[AFTERNTH₂, active₁]

Status:

AFTERNTH₂: [2,1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(80) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(81) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(82) TRUE

(83) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(84) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

mark₁ > CONS₂

Status:

CONS₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(85) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, active(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x2)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > CONS₁

Status:

CONS₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(87) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(active(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(active(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[CONS₂, active₁]

Status:

CONS₂: [2,1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(89) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(90) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(91) TRUE

(92) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PAIR(X1, mark(X2)) → PAIR(X1, X2)
PAIR(mark(X1), X2) → PAIR(X1, X2)
PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(93) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(X1, mark(X2)) → PAIR(X1, X2)
PAIR(mark(X1), X2) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PAIR(x1, x2)  =  PAIR(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

mark₁ > PAIR₂

Status:

PAIR₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(94) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(X1, active(X2)) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PAIR(x1, x2)  =  PAIR(x2)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > PAIR₁

Status:

PAIR₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(96) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PAIR(active(X1), X2) → PAIR(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(97) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(active(X1), X2) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[PAIR₂, active₁]

Status:

PAIR₂: [2,1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(98) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(99) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(100) TRUE

(101) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)
SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)
SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(102) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)
SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SPLITAT(x1, x2)  =  SPLITAT(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

mark₁ > SPLITAT₂

Status:

SPLITAT₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(103) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(104) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SPLITAT(x1, x2)  =  SPLITAT(x2)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > SPLITAT₁

Status:

SPLITAT₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(105) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SPLITAT(active(X1), X2) → SPLITAT(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(106) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SPLITAT(active(X1), X2) → SPLITAT(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[SPLITAT₂, active₁]

Status:

SPLITAT₂: [2,1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(107) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(108) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(109) TRUE

(110) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12¹(X1, mark(X2)) → U12¹(X1, X2)
U12¹(mark(X1), X2) → U12¹(X1, X2)
U12¹(active(X1), X2) → U12¹(X1, X2)
U12¹(X1, active(X2)) → U12¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(111) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, mark(X2)) → U12¹(X1, X2)
U12¹(mark(X1), X2) → U12¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2)  =  U12¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

mark₁ > U12^1₂

Status:

U12^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(112) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12¹(active(X1), X2) → U12¹(X1, X2)
U12¹(X1, active(X2)) → U12¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(113) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, active(X2)) → U12¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2)  =  U12¹(x2)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > U12^1₁

Status:

U12^1₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(114) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12¹(active(X1), X2) → U12¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(115) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(active(X1), X2) → U12¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[U12^1₂, active₁]

Status:

U12^1₂: [2,1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(116) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(117) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(118) TRUE

(119) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(X1, mark(X2), X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(mark(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, X2, mark(X3), X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, X2, X3, mark(X4)) → U11¹(X1, X2, X3, X4)
U11¹(active(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, active(X2), X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, X2, active(X3), X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, X2, X3, active(X4)) → U11¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(120) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, X2, X3, mark(X4)) → U11¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3, x4)  =  U11¹(x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U11^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(121) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(X1, mark(X2), X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(mark(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, X2, mark(X3), X4) → U11¹(X1, X2, X3, X4)
U11¹(active(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, active(X2), X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, X2, active(X3), X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, X2, X3, active(X4)) → U11¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(122) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, X2, mark(X3), X4) → U11¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3, x4)  =  U11¹(x3, x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U11^1₂, mark₁]

Status:

U11^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(123) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(X1, mark(X2), X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(mark(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(active(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, active(X2), X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, X2, active(X3), X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, X2, X3, active(X4)) → U11¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(124) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, X2, X3, active(X4)) → U11¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3, x4)  =  x4
mark(x1)  =  mark
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark: []
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(125) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(X1, mark(X2), X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(mark(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(active(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, active(X2), X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, X2, active(X3), X4) → U11¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(126) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, X2, active(X3), X4) → U11¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3, x4)  =  U11¹(x3, x4)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U11^1₂: [2,1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(127) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(X1, mark(X2), X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(mark(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(active(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, active(X2), X3, X4) → U11¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(128) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, mark(X2), X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(X1, active(X2), X3, X4) → U11¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3, x4)  =  U11¹(x2, x4)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

mark₁ > [U11^1₂, active₁]

Status:

U11^1₂: [1,2]
mark₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(129) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(mark(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
U11¹(active(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(130) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(mark(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3, x4)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(131) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(active(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(132) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(active(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3, x4)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(133) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(134) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(135) TRUE

(136) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2, X3, X4)) → ACTIVE(U11(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, N, X, XS)) → MARK(U12(splitAt(N, XS), X))
MARK(U11(X1, X2, X3, X4)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
ACTIVE(U12(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(U12(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
ACTIVE(afterNth(N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(pair(X1, X2)) → ACTIVE(pair(mark(X1), mark(X2)))
ACTIVE(and(tt, X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(fst(pair(X, Y))) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(snd(X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(sel(N, XS)) → MARK(head(afterNth(N, XS)))
MARK(and(X1, X2)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
ACTIVE(snd(pair(X, Y))) → MARK(Y)
MARK(fst(X)) → MARK(X)
MARK(head(X)) → ACTIVE(head(mark(X)))
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(head(X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U11(tt, N, X, XS))
MARK(natsFrom(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(tail(cons(N, XS))) → MARK(XS)
MARK(s(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(take(N, XS)) → MARK(fst(splitAt(N, XS)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(137) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(pair(X1, X2)) → ACTIVE(pair(mark(X1), mark(X2)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
U11(x1, x2, x3, x4)  =  U11
ACTIVE(x1)  =  x1
mark(x1)  =  mark(x1)
tt  =  tt
U12(x1, x2)  =  U12
splitAt(x1, x2)  =  splitAt
pair(x1, x2)  =  pair
cons(x1, x2)  =  cons
afterNth(x1, x2)  =  afterNth
snd(x1)  =  snd
and(x1, x2)  =  and
fst(x1)  =  fst
head(x1)  =  head
natsFrom(x1)  =  natsFrom
s(x1)  =  s
sel(x1, x2)  =  sel
0  =  0
nil  =  nil
tail(x1)  =  tail
take(x1, x2)  =  take
active(x1)  =  active

Lexicographic path order with status [LPO].
Quasi-Precedence:

mark₁ > 0
mark₁ > active > [MARK, U11, U12, splitAt, cons, afterNth, snd, and, fst, head, natsFrom, s, sel, tail, take] > tt
mark₁ > active > [MARK, U11, U12, splitAt, cons, afterNth, snd, and, fst, head, natsFrom, s, sel, tail, take] > pair
nil > active > [MARK, U11, U12, splitAt, cons, afterNth, snd, and, fst, head, natsFrom, s, sel, tail, take] > tt
nil > active > [MARK, U11, U12, splitAt, cons, afterNth, snd, and, fst, head, natsFrom, s, sel, tail, take] > pair

Status:

MARK: []
U11: []
mark₁: [1]
tt: []
U12: []
splitAt: []
pair: []
cons: []
afterNth: []
snd: []
and: []
fst: []
head: []
natsFrom: []
s: []
sel: []
0: []
nil: []
tail: []
take: []
active: []

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
snd(active(X)) → snd(X)
snd(mark(X)) → snd(X)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
natsFrom(active(X)) → natsFrom(X)
natsFrom(mark(X)) → natsFrom(X)
and(X1, mark(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
head(active(X)) → head(X)
head(mark(X)) → head(X)
fst(active(X)) → fst(X)
fst(mark(X)) → fst(X)
sel(X1, mark(X2)) → sel(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(active(X)) → tail(X)
tail(mark(X)) → tail(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(138) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2, X3, X4)) → ACTIVE(U11(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, N, X, XS)) → MARK(U12(splitAt(N, XS), X))
MARK(U11(X1, X2, X3, X4)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
ACTIVE(U12(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(U12(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
ACTIVE(afterNth(N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
ACTIVE(and(tt, X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(fst(pair(X, Y))) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(snd(X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(sel(N, XS)) → MARK(head(afterNth(N, XS)))
MARK(and(X1, X2)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
ACTIVE(snd(pair(X, Y))) → MARK(Y)
MARK(fst(X)) → MARK(X)
MARK(head(X)) → ACTIVE(head(mark(X)))
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(head(X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U11(tt, N, X, XS))
MARK(natsFrom(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(tail(cons(N, XS))) → MARK(XS)
MARK(s(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(take(N, XS)) → MARK(fst(splitAt(N, XS)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(139) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(s(X)) → ACTIVE(s(mark(X)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
U11(x1, x2, x3, x4)  =  U11
ACTIVE(x1)  =  x1
mark(x1)  =  x1
tt  =  tt
U12(x1, x2)  =  U12
splitAt(x1, x2)  =  splitAt
pair(x1, x2)  =  pair
cons(x1, x2)  =  cons
afterNth(x1, x2)  =  afterNth
snd(x1)  =  snd
and(x1, x2)  =  and
fst(x1)  =  fst
head(x1)  =  head
natsFrom(x1)  =  natsFrom
s(x1)  =  s
sel(x1, x2)  =  sel
0  =  0
nil  =  nil
tail(x1)  =  tail
take(x1, x2)  =  take
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

pair > [MARK, U11, tt, U12, splitAt, cons, afterNth, snd, and, fst, head, natsFrom, sel, tail, take, active₁] > [s, nil]
0 > [s, nil]

Status:

MARK: []
U11: []
tt: []
U12: []
splitAt: []
pair: []
cons: []
afterNth: []
snd: []
and: []
fst: []
head: []
natsFrom: []
s: []
sel: []
0: []
nil: []
tail: []
take: []
active₁: [1]

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
snd(active(X)) → snd(X)
snd(mark(X)) → snd(X)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
natsFrom(active(X)) → natsFrom(X)
natsFrom(mark(X)) → natsFrom(X)
and(X1, mark(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
head(active(X)) → head(X)
head(mark(X)) → head(X)
fst(active(X)) → fst(X)
fst(mark(X)) → fst(X)
sel(X1, mark(X2)) → sel(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(active(X)) → tail(X)
tail(mark(X)) → tail(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(140) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2, X3, X4)) → ACTIVE(U11(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, N, X, XS)) → MARK(U12(splitAt(N, XS), X))
MARK(U11(X1, X2, X3, X4)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
ACTIVE(U12(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(U12(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
ACTIVE(afterNth(N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
ACTIVE(and(tt, X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(fst(pair(X, Y))) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(snd(X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(sel(N, XS)) → MARK(head(afterNth(N, XS)))
MARK(and(X1, X2)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
ACTIVE(snd(pair(X, Y))) → MARK(Y)
MARK(fst(X)) → MARK(X)
MARK(head(X)) → ACTIVE(head(mark(X)))
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(head(X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U11(tt, N, X, XS))
MARK(natsFrom(X)) → MARK(X)
ACTIVE(tail(cons(N, XS))) → MARK(XS)
MARK(s(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(take(N, XS)) → MARK(fst(splitAt(N, XS)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(141) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
U11(x1, x2, x3, x4)  =  U11
ACTIVE(x1)  =  x1
mark(x1)  =  mark
tt  =  tt
U12(x1, x2)  =  U12
splitAt(x1, x2)  =  splitAt
pair(x1, x2)  =  x1
cons(x1, x2)  =  cons
afterNth(x1, x2)  =  afterNth
snd(x1)  =  snd
and(x1, x2)  =  and
fst(x1)  =  fst
head(x1)  =  head
natsFrom(x1)  =  natsFrom
s(x1)  =  s(x1)
sel(x1, x2)  =  sel
0  =  0
nil  =  nil
tail(x1)  =  tail
take(x1, x2)  =  take
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[MARK, U11, mark, U12, splitAt, afterNth, snd, and, fst, head, natsFrom, sel, nil, tail, take] > cons > [tt, s₁]
[MARK, U11, mark, U12, splitAt, afterNth, snd, and, fst, head, natsFrom, sel, nil, tail, take] > 0

Status:

MARK: []
U11: []
mark: []
tt: []
U12: []
splitAt: []
cons: []
afterNth: []
snd: []
and: []
fst: []
head: []
natsFrom: []
s₁: [1]
sel: []
0: []
nil: []
tail: []
take: []

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
snd(active(X)) → snd(X)
snd(mark(X)) → snd(X)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
natsFrom(active(X)) → natsFrom(X)
natsFrom(mark(X)) → natsFrom(X)
and(X1, mark(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
head(active(X)) → head(X)
head(mark(X)) → head(X)
fst(active(X)) → fst(X)
fst(mark(X)) → fst(X)
sel(X1, mark(X2)) → sel(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(active(X)) → tail(X)
tail(mark(X)) → tail(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(142) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2, X3, X4)) → ACTIVE(U11(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, N, X, XS)) → MARK(U12(splitAt(N, XS), X))
MARK(U11(X1, X2, X3, X4)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
ACTIVE(U12(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(U12(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
ACTIVE(afterNth(N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
ACTIVE(and(tt, X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
ACTIVE(fst(pair(X, Y))) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(snd(X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(sel(N, XS)) → MARK(head(afterNth(N, XS)))
MARK(and(X1, X2)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
ACTIVE(snd(pair(X, Y))) → MARK(Y)
MARK(fst(X)) → MARK(X)
MARK(head(X)) → ACTIVE(head(mark(X)))
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(head(X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U11(tt, N, X, XS))
MARK(natsFrom(X)) → MARK(X)
ACTIVE(tail(cons(N, XS))) → MARK(XS)
MARK(s(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(take(N, XS)) → MARK(fst(splitAt(N, XS)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
mark(U11(X1, X2, X3, X4)) → active(U11(mark(X1), X2, X3, X4))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(fst(X)) → active(fst(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, mark(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, mark(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, mark(X4)) → U11(X1, X2, X3, X4)
U11(active(X1), X2, X3, X4) → U11(X1, X2, X3, X4)
U11(X1, active(X2), X3, X4) → U11(X1, X2, X3, X4)
U11(X1, X2, active(X3), X4) → U11(X1, X2, X3, X4)
U11(X1, X2, X3, active(X4)) → U11(X1, X2, X3, X4)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.