Termination w.r.t. Q proof of /tmp/tmptprP-Z/LISTUTILITIES_nosorts

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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.

↳ QTRS

  ↳ DependencyPairsProof

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.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

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

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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

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

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.
The approximation of the Dependency Graph [15,17,22] contains 16 SCCs with 33 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

TAKE(X1, active(X2)) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

TAKE(X1, active(X2)) → TAKE(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
TAKE(active(X1), X2) → TAKE(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
TAKE(mark(X1), X2) → TAKE(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
TAKE(X1, mark(X2)) → TAKE(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

TAIL(active(X)) → TAIL(X)
The graph contains the following edges 1 > 1
TAIL(mark(X)) → TAIL(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

SEL(mark(X1), X2) → SEL(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
SEL(X1, active(X2)) → SEL(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
SEL(X1, mark(X2)) → SEL(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
SEL(active(X1), X2) → SEL(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S(mark(X)) → S(X)
S(active(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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

From the DPs we obtained the following set of size-change graphs:

S(mark(X)) → S(X)
The graph contains the following edges 1 > 1
S(active(X)) → S(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

NATSFROM(active(X)) → NATSFROM(X)
The graph contains the following edges 1 > 1
NATSFROM(mark(X)) → NATSFROM(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

HEAD(mark(X)) → HEAD(X)
HEAD(active(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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

HEAD(mark(X)) → HEAD(X)
The graph contains the following edges 1 > 1
HEAD(active(X)) → HEAD(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

FST(active(X)) → FST(X)
The graph contains the following edges 1 > 1
FST(mark(X)) → FST(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

AND(mark(X1), X2) → AND(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
AND(active(X1), X2) → AND(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
AND(X1, mark(X2)) → AND(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
AND(X1, active(X2)) → AND(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

SND(mark(X)) → SND(X)
SND(active(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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

SND(mark(X)) → SND(X)
The graph contains the following edges 1 > 1
SND(active(X)) → SND(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

From the DPs we obtained the following set of size-change graphs:

AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

CONS(X1, active(X2)) → CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
CONS(mark(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS(active(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS(X1, mark(X2)) → CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

PAIR(X1, active(X2)) → PAIR(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
PAIR(mark(X1), X2) → PAIR(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
PAIR(X1, mark(X2)) → PAIR(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
PAIR(active(X1), X2) → PAIR(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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(X1, active(X2)) → SPLITAT(X1, X2)
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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

From the DPs we obtained the following set of size-change graphs:

SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

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(X1, active(X2)) → U12¹(X1, X2)
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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

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)

From the DPs we obtained the following set of size-change graphs:

U12¹(X1, mark(X2)) → U12¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U12¹(mark(X1), X2) → U12¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U12¹(X1, active(X2)) → U12¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U12¹(active(X1), X2) → U12¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

U11¹(active(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
U11¹(mark(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
U11¹(X1, active(X2), X3, X4) → U11¹(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
U11¹(X1, X2, X3, active(X4)) → U11¹(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4
U11¹(X1, mark(X2), X3, X4) → U11¹(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
U11¹(X1, X2, active(X3), X4) → U11¹(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
U11¹(X1, X2, X3, mark(X4)) → U11¹(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4
U11¹(X1, X2, mark(X3), X4) → U11¹(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(pair(X1, X2)) → ACTIVE(pair(mark(X1), mark(X2)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃, x₄)) = 1
POL(U12(x₁, x₂)) = 1
POL(active(x₁)) = 0
POL(afterNth(x₁, x₂)) = 1
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(fst(x₁)) = 1
POL(head(x₁)) = 1
POL(mark(x₁)) = 0
POL(natsFrom(x₁)) = 1
POL(nil) = 0
POL(pair(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(sel(x₁, x₂)) = 1
POL(snd(x₁)) = 1
POL(splitAt(x₁, x₂)) = 1
POL(tail(x₁)) = 1
POL(take(x₁, x₂)) = 1
POL(tt) = 0

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

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

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

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.
By narrowing [15] the rule MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2)) at position [0] we obtained the following new rules:

MARK(U12(sel(x0, x1), y1)) → ACTIVE(U12(active(sel(mark(x0), mark(x1))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(U12(fst(x0), y1)) → ACTIVE(U12(active(fst(mark(x0))), y1))
MARK(U12(pair(x0, x1), y1)) → ACTIVE(U12(active(pair(mark(x0), mark(x1))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U12(afterNth(x0, x1), y1)) → ACTIVE(U12(active(afterNth(mark(x0), mark(x1))), y1))
MARK(U12(natsFrom(x0), y1)) → ACTIVE(U12(active(natsFrom(mark(x0))), y1))
MARK(U12(head(x0), y1)) → ACTIVE(U12(active(head(mark(x0))), y1))
MARK(U12(splitAt(x0, x1), y1)) → ACTIVE(U12(active(splitAt(mark(x0), mark(x1))), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U12(snd(x0), y1)) → ACTIVE(U12(active(snd(mark(x0))), y1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(U12(and(x0, x1), y1)) → ACTIVE(U12(active(and(mark(x0), x1)), y1))
MARK(U12(tail(x0), y1)) → ACTIVE(U12(active(tail(mark(x0))), y1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U11(mark(x0), x1, x2, x3)), y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

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

MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(head(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(U12(afterNth(x0, x1), y1)) → ACTIVE(U12(active(afterNth(mark(x0), mark(x1))), y1))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U11(tt, N, X, XS))
ACTIVE(afterNth(N, XS)) → MARK(snd(splitAt(N, XS)))
ACTIVE(take(N, XS)) → MARK(fst(splitAt(N, XS)))
ACTIVE(sel(N, XS)) → MARK(head(afterNth(N, XS)))
MARK(take(X1, X2)) → MARK(X1)
MARK(U12(and(x0, x1), y1)) → ACTIVE(U12(active(and(mark(x0), x1)), y1))
ACTIVE(snd(pair(X, Y))) → MARK(Y)
MARK(fst(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U12(pair(x0, x1), y1)) → ACTIVE(U12(active(pair(mark(x0), mark(x1))), y1))
MARK(U11(X1, X2, X3, X4)) → ACTIVE(U11(mark(X1), X2, X3, X4))
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X1)
MARK(U12(snd(x0), y1)) → ACTIVE(U12(active(snd(mark(x0))), y1))
ACTIVE(fst(pair(X, Y))) → MARK(X)
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(U12(U11(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(afterNth(X1, X2)) → MARK(X1)
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(U12(sel(x0, x1), y1)) → ACTIVE(U12(active(sel(mark(x0), mark(x1))), y1))
MARK(take(X1, X2)) → MARK(X2)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(fst(x0), y1)) → ACTIVE(U12(active(fst(mark(x0))), y1))
MARK(s(X)) → MARK(X)
MARK(U12(head(x0), y1)) → ACTIVE(U12(active(head(mark(x0))), y1))
MARK(natsFrom(X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(U12(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
ACTIVE(U11(tt, N, X, XS)) → MARK(U12(splitAt(N, XS), X))
MARK(pair(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(snd(X)) → MARK(X)
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(U12(natsFrom(x0), y1)) → ACTIVE(U12(active(natsFrom(mark(x0))), y1))
MARK(U12(splitAt(x0, x1), y1)) → ACTIVE(U12(active(splitAt(mark(x0), mark(x1))), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(U12(tail(x0), y1)) → ACTIVE(U12(active(tail(mark(x0))), y1))
MARK(U11(X1, X2, X3, X4)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(sel(X1, X2)) → MARK(X2)
ACTIVE(tail(cons(N, XS))) → MARK(XS)

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.
By narrowing [15] the rule MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2)) at position [0] we obtained the following new rules:

MARK(and(y0, active(x1))) → ACTIVE(and(mark(y0), x1))
MARK(and(sel(x0, x1), y1)) → ACTIVE(and(active(sel(mark(x0), mark(x1))), y1))
MARK(and(natsFrom(x0), y1)) → ACTIVE(and(active(natsFrom(mark(x0))), y1))
MARK(and(snd(x0), y1)) → ACTIVE(and(active(snd(mark(x0))), y1))
MARK(and(take(x0, x1), y1)) → ACTIVE(and(active(take(mark(x0), mark(x1))), y1))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), y1))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), x1)), y1))
MARK(and(pair(x0, x1), y1)) → ACTIVE(and(active(pair(mark(x0), mark(x1))), y1))
MARK(and(afterNth(x0, x1), y1)) → ACTIVE(and(active(afterNth(mark(x0), mark(x1))), y1))
MARK(and(U12(x0, x1), y1)) → ACTIVE(and(active(U12(mark(x0), x1)), y1))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), y1))
MARK(and(0, y1)) → ACTIVE(and(active(0), y1))
MARK(and(x0, x1)) → ACTIVE(and(x0, x1))
MARK(and(y0, mark(x1))) → ACTIVE(and(mark(y0), x1))
MARK(and(fst(x0), y1)) → ACTIVE(and(active(fst(mark(x0))), y1))
MARK(and(nil, y1)) → ACTIVE(and(active(nil), y1))
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), y1))
MARK(and(head(x0), y1)) → ACTIVE(and(active(head(mark(x0))), y1))
MARK(and(splitAt(x0, x1), y1)) → ACTIVE(and(active(splitAt(mark(x0), mark(x1))), y1))
MARK(and(U11(x0, x1, x2, x3), y1)) → ACTIVE(and(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(and(tail(x0), y1)) → ACTIVE(and(active(tail(mark(x0))), y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

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

MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(tail(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), x1)), y1))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), y1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U12(afterNth(x0, x1), y1)) → ACTIVE(U12(active(afterNth(mark(x0), mark(x1))), y1))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U11(tt, N, X, XS))
MARK(and(pair(x0, x1), y1)) → ACTIVE(and(active(pair(mark(x0), mark(x1))), y1))
ACTIVE(afterNth(N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(and(U12(x0, x1), y1)) → ACTIVE(and(active(U12(mark(x0), x1)), y1))
ACTIVE(take(N, XS)) → MARK(fst(splitAt(N, XS)))
ACTIVE(sel(N, XS)) → MARK(head(afterNth(N, XS)))
MARK(and(nil, y1)) → ACTIVE(and(active(nil), y1))
MARK(take(X1, X2)) → MARK(X1)
ACTIVE(snd(pair(X, Y))) → MARK(Y)
MARK(U12(and(x0, x1), y1)) → ACTIVE(U12(active(and(mark(x0), x1)), y1))
MARK(fst(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(and(y0, active(x1))) → ACTIVE(and(mark(y0), x1))
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U12(pair(x0, x1), y1)) → ACTIVE(U12(active(pair(mark(x0), mark(x1))), y1))
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(U11(X1, X2, X3, X4)) → ACTIVE(U11(mark(X1), X2, X3, X4))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(pair(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(and(afterNth(x0, x1), y1)) → ACTIVE(and(active(afterNth(mark(x0), mark(x1))), y1))
MARK(U12(snd(x0), y1)) → ACTIVE(U12(active(snd(mark(x0))), y1))
MARK(and(x0, x1)) → ACTIVE(and(x0, x1))
ACTIVE(fst(pair(X, Y))) → MARK(X)
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
ACTIVE(and(tt, X)) → MARK(X)
MARK(and(U11(x0, x1, x2, x3), y1)) → ACTIVE(and(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(and(tail(x0), y1)) → ACTIVE(and(active(tail(mark(x0))), y1))
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(U12(U11(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(U12(sel(x0, x1), y1)) → ACTIVE(U12(active(sel(mark(x0), mark(x1))), y1))
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(take(X1, X2)) → MARK(X2)
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
MARK(U12(fst(x0), y1)) → ACTIVE(U12(active(fst(mark(x0))), y1))
MARK(s(X)) → MARK(X)
MARK(and(snd(x0), y1)) → ACTIVE(and(active(snd(mark(x0))), y1))
MARK(and(natsFrom(x0), y1)) → ACTIVE(and(active(natsFrom(mark(x0))), y1))
MARK(and(sel(x0, x1), y1)) → ACTIVE(and(active(sel(mark(x0), mark(x1))), y1))
MARK(and(take(x0, x1), y1)) → ACTIVE(and(active(take(mark(x0), mark(x1))), y1))
MARK(U12(head(x0), y1)) → ACTIVE(U12(active(head(mark(x0))), y1))
MARK(natsFrom(X)) → MARK(X)
ACTIVE(U12(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(and(0, y1)) → ACTIVE(and(active(0), y1))
MARK(and(y0, mark(x1))) → ACTIVE(and(mark(y0), x1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
ACTIVE(U11(tt, N, X, XS)) → MARK(U12(splitAt(N, XS), X))
MARK(pair(X1, X2)) → MARK(X2)
MARK(and(fst(x0), y1)) → ACTIVE(and(active(fst(mark(x0))), y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
MARK(and(splitAt(x0, x1), y1)) → ACTIVE(and(active(splitAt(mark(x0), mark(x1))), y1))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(snd(X)) → MARK(X)
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(U12(natsFrom(x0), y1)) → ACTIVE(U12(active(natsFrom(mark(x0))), y1))
MARK(U12(splitAt(x0, x1), y1)) → ACTIVE(U12(active(splitAt(mark(x0), mark(x1))), y1))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), y1))
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(and(head(x0), y1)) → ACTIVE(and(active(head(mark(x0))), y1))
MARK(U12(tail(x0), y1)) → ACTIVE(U12(active(tail(mark(x0))), y1))
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(U11(X1, X2, X3, X4)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
ACTIVE(tail(cons(N, XS))) → MARK(XS)

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.
By narrowing [15] the rule MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2))) at position [0] we obtained the following new rules:

MARK(splitAt(tt, y1)) → ACTIVE(splitAt(active(tt), mark(y1)))
MARK(splitAt(y0, tt)) → ACTIVE(splitAt(mark(y0), active(tt)))
MARK(splitAt(y0, U12(x0, x1))) → ACTIVE(splitAt(mark(y0), active(U12(mark(x0), x1))))
MARK(splitAt(x0, y1)) → ACTIVE(splitAt(x0, mark(y1)))
MARK(splitAt(fst(x0), y1)) → ACTIVE(splitAt(active(fst(mark(x0))), mark(y1)))
MARK(splitAt(and(x0, x1), y1)) → ACTIVE(splitAt(active(and(mark(x0), x1)), mark(y1)))
MARK(splitAt(y0, pair(x0, x1))) → ACTIVE(splitAt(mark(y0), active(pair(mark(x0), mark(x1)))))
MARK(splitAt(y0, sel(x0, x1))) → ACTIVE(splitAt(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(splitAt(natsFrom(x0), y1)) → ACTIVE(splitAt(active(natsFrom(mark(x0))), mark(y1)))
MARK(splitAt(y0, natsFrom(x0))) → ACTIVE(splitAt(mark(y0), active(natsFrom(mark(x0)))))
MARK(splitAt(y0, tail(x0))) → ACTIVE(splitAt(mark(y0), active(tail(mark(x0)))))
MARK(splitAt(y0, head(x0))) → ACTIVE(splitAt(mark(y0), active(head(mark(x0)))))
MARK(splitAt(U11(x0, x1, x2, x3), y1)) → ACTIVE(splitAt(active(U11(mark(x0), x1, x2, x3)), mark(y1)))
MARK(splitAt(y0, nil)) → ACTIVE(splitAt(mark(y0), active(nil)))
MARK(splitAt(y0, s(x0))) → ACTIVE(splitAt(mark(y0), active(s(mark(x0)))))
MARK(splitAt(pair(x0, x1), y1)) → ACTIVE(splitAt(active(pair(mark(x0), mark(x1))), mark(y1)))
MARK(splitAt(nil, y1)) → ACTIVE(splitAt(active(nil), mark(y1)))
MARK(splitAt(tail(x0), y1)) → ACTIVE(splitAt(active(tail(mark(x0))), mark(y1)))
MARK(splitAt(y0, take(x0, x1))) → ACTIVE(splitAt(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(splitAt(take(x0, x1), y1)) → ACTIVE(splitAt(active(take(mark(x0), mark(x1))), mark(y1)))
MARK(splitAt(U12(x0, x1), y1)) → ACTIVE(splitAt(active(U12(mark(x0), x1)), mark(y1)))
MARK(splitAt(cons(x0, x1), y1)) → ACTIVE(splitAt(active(cons(mark(x0), x1)), mark(y1)))
MARK(splitAt(y0, x1)) → ACTIVE(splitAt(mark(y0), x1))
MARK(splitAt(y0, splitAt(x0, x1))) → ACTIVE(splitAt(mark(y0), active(splitAt(mark(x0), mark(x1)))))
MARK(splitAt(s(x0), y1)) → ACTIVE(splitAt(active(s(mark(x0))), mark(y1)))
MARK(splitAt(sel(x0, x1), y1)) → ACTIVE(splitAt(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(splitAt(y0, U11(x0, x1, x2, x3))) → ACTIVE(splitAt(mark(y0), active(U11(mark(x0), x1, x2, x3))))
MARK(splitAt(y0, snd(x0))) → ACTIVE(splitAt(mark(y0), active(snd(mark(x0)))))
MARK(splitAt(afterNth(x0, x1), y1)) → ACTIVE(splitAt(active(afterNth(mark(x0), mark(x1))), mark(y1)))
MARK(splitAt(splitAt(x0, x1), y1)) → ACTIVE(splitAt(active(splitAt(mark(x0), mark(x1))), mark(y1)))
MARK(splitAt(y0, afterNth(x0, x1))) → ACTIVE(splitAt(mark(y0), active(afterNth(mark(x0), mark(x1)))))
MARK(splitAt(head(x0), y1)) → ACTIVE(splitAt(active(head(mark(x0))), mark(y1)))
MARK(splitAt(y0, cons(x0, x1))) → ACTIVE(splitAt(mark(y0), active(cons(mark(x0), x1))))
MARK(splitAt(snd(x0), y1)) → ACTIVE(splitAt(active(snd(mark(x0))), mark(y1)))
MARK(splitAt(y0, and(x0, x1))) → ACTIVE(splitAt(mark(y0), active(and(mark(x0), x1))))
MARK(splitAt(y0, fst(x0))) → ACTIVE(splitAt(mark(y0), active(fst(mark(x0)))))
MARK(splitAt(y0, 0)) → ACTIVE(splitAt(mark(y0), active(0)))
MARK(splitAt(0, y1)) → ACTIVE(splitAt(active(0), mark(y1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

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

MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(splitAt(y0, U12(x0, x1))) → ACTIVE(splitAt(mark(y0), active(U12(mark(x0), x1))))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), x1)), y1))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), y1))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U11(tt, N, X, XS))
MARK(and(pair(x0, x1), y1)) → ACTIVE(and(active(pair(mark(x0), mark(x1))), y1))
ACTIVE(take(N, XS)) → MARK(fst(splitAt(N, XS)))
ACTIVE(sel(N, XS)) → MARK(head(afterNth(N, XS)))
MARK(splitAt(natsFrom(x0), y1)) → ACTIVE(splitAt(active(natsFrom(mark(x0))), mark(y1)))
MARK(and(nil, y1)) → ACTIVE(and(active(nil), y1))
MARK(fst(X)) → MARK(X)
MARK(splitAt(y0, take(x0, x1))) → ACTIVE(splitAt(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U12(pair(x0, x1), y1)) → ACTIVE(U12(active(pair(mark(x0), mark(x1))), y1))
MARK(splitAt(y0, splitAt(x0, x1))) → ACTIVE(splitAt(mark(y0), active(splitAt(mark(x0), mark(x1)))))
MARK(splitAt(sel(x0, x1), y1)) → ACTIVE(splitAt(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(splitAt(y0, U11(x0, x1, x2, x3))) → ACTIVE(splitAt(mark(y0), active(U11(mark(x0), x1, x2, x3))))
MARK(splitAt(y0, snd(x0))) → ACTIVE(splitAt(mark(y0), active(snd(mark(x0)))))
MARK(and(afterNth(x0, x1), y1)) → ACTIVE(and(active(afterNth(mark(x0), mark(x1))), y1))
MARK(splitAt(y0, afterNth(x0, x1))) → ACTIVE(splitAt(mark(y0), active(afterNth(mark(x0), mark(x1)))))
MARK(and(x0, x1)) → ACTIVE(and(x0, x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(splitAt(y0, and(x0, x1))) → ACTIVE(splitAt(mark(y0), active(and(mark(x0), x1))))
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
ACTIVE(and(tt, X)) → MARK(X)
MARK(and(U11(x0, x1, x2, x3), y1)) → ACTIVE(and(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(U12(U11(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(U12(sel(x0, x1), y1)) → ACTIVE(U12(active(sel(mark(x0), mark(x1))), y1))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(splitAt(tt, y1)) → ACTIVE(splitAt(active(tt), mark(y1)))
MARK(splitAt(y0, tt)) → ACTIVE(splitAt(mark(y0), active(tt)))
MARK(take(X1, X2)) → MARK(X2)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
MARK(U12(fst(x0), y1)) → ACTIVE(U12(active(fst(mark(x0))), y1))
MARK(s(X)) → MARK(X)
MARK(and(snd(x0), y1)) → ACTIVE(and(active(snd(mark(x0))), y1))
MARK(and(sel(x0, x1), y1)) → ACTIVE(and(active(sel(mark(x0), mark(x1))), y1))
MARK(splitAt(y0, pair(x0, x1))) → ACTIVE(splitAt(mark(y0), active(pair(mark(x0), mark(x1)))))
ACTIVE(U12(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(and(0, y1)) → ACTIVE(and(active(0), y1))
MARK(and(y0, mark(x1))) → ACTIVE(and(mark(y0), x1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
ACTIVE(U11(tt, N, X, XS)) → MARK(U12(splitAt(N, XS), X))
MARK(and(X1, X2)) → MARK(X1)
MARK(splitAt(y0, tail(x0))) → ACTIVE(splitAt(mark(y0), active(tail(mark(x0)))))
MARK(splitAt(y0, head(x0))) → ACTIVE(splitAt(mark(y0), active(head(mark(x0)))))
MARK(splitAt(U11(x0, x1, x2, x3), y1)) → ACTIVE(splitAt(active(U11(mark(x0), x1, x2, x3)), mark(y1)))
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(splitAt(nil, y1)) → ACTIVE(splitAt(active(nil), mark(y1)))
MARK(splitAt(y0, nil)) → ACTIVE(splitAt(mark(y0), active(nil)))
MARK(splitAt(y0, x1)) → ACTIVE(splitAt(mark(y0), x1))
MARK(U12(natsFrom(x0), y1)) → ACTIVE(U12(active(natsFrom(mark(x0))), y1))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(splitAt(y0, cons(x0, x1))) → ACTIVE(splitAt(mark(y0), active(cons(mark(x0), x1))))
MARK(splitAt(snd(x0), y1)) → ACTIVE(splitAt(active(snd(mark(x0))), mark(y1)))
MARK(and(head(x0), y1)) → ACTIVE(and(active(head(mark(x0))), y1))
MARK(U12(tail(x0), y1)) → ACTIVE(U12(active(tail(mark(x0))), y1))
MARK(U11(X1, X2, X3, X4)) → MARK(X1)
MARK(splitAt(0, y1)) → ACTIVE(splitAt(active(0), mark(y1)))
MARK(splitAt(y0, 0)) → ACTIVE(splitAt(mark(y0), active(0)))
MARK(sel(X1, X2)) → MARK(X2)
ACTIVE(tail(cons(N, XS))) → MARK(XS)
MARK(tail(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U12(afterNth(x0, x1), y1)) → ACTIVE(U12(active(afterNth(mark(x0), mark(x1))), y1))
MARK(splitAt(and(x0, x1), y1)) → ACTIVE(splitAt(active(and(mark(x0), x1)), mark(y1)))
ACTIVE(afterNth(N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(and(U12(x0, x1), y1)) → ACTIVE(and(active(U12(mark(x0), x1)), y1))
MARK(splitAt(y0, natsFrom(x0))) → ACTIVE(splitAt(mark(y0), active(natsFrom(mark(x0)))))
MARK(take(X1, X2)) → MARK(X1)
ACTIVE(snd(pair(X, Y))) → MARK(Y)
MARK(U12(and(x0, x1), y1)) → ACTIVE(U12(active(and(mark(x0), x1)), y1))
MARK(splitAt(pair(x0, x1), y1)) → ACTIVE(splitAt(active(pair(mark(x0), mark(x1))), mark(y1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(splitAt(take(x0, x1), y1)) → ACTIVE(splitAt(active(take(mark(x0), mark(x1))), mark(y1)))
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(and(y0, active(x1))) → ACTIVE(and(mark(y0), x1))
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(splitAt(U12(x0, x1), y1)) → ACTIVE(splitAt(active(U12(mark(x0), x1)), mark(y1)))
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(U11(X1, X2, X3, X4)) → ACTIVE(U11(mark(X1), X2, X3, X4))
MARK(pair(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(U12(snd(x0), y1)) → ACTIVE(U12(active(snd(mark(x0))), y1))
ACTIVE(fst(pair(X, Y))) → MARK(X)
MARK(splitAt(y0, fst(x0))) → ACTIVE(splitAt(mark(y0), active(fst(mark(x0)))))
MARK(and(tail(x0), y1)) → ACTIVE(and(active(tail(mark(x0))), y1))
MARK(afterNth(X1, X2)) → MARK(X1)
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(splitAt(x0, y1)) → ACTIVE(splitAt(x0, mark(y1)))
MARK(and(natsFrom(x0), y1)) → ACTIVE(and(active(natsFrom(mark(x0))), y1))
MARK(splitAt(fst(x0), y1)) → ACTIVE(splitAt(active(fst(mark(x0))), mark(y1)))
MARK(and(take(x0, x1), y1)) → ACTIVE(and(active(take(mark(x0), mark(x1))), y1))
MARK(U12(head(x0), y1)) → ACTIVE(U12(active(head(mark(x0))), y1))
MARK(natsFrom(X)) → MARK(X)
MARK(splitAt(y0, sel(x0, x1))) → ACTIVE(splitAt(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(pair(X1, X2)) → MARK(X2)
MARK(and(fst(x0), y1)) → ACTIVE(and(active(fst(mark(x0))), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(splitAt(y0, s(x0))) → ACTIVE(splitAt(mark(y0), active(s(mark(x0)))))
MARK(splitAt(tail(x0), y1)) → ACTIVE(splitAt(active(tail(mark(x0))), mark(y1)))
MARK(and(splitAt(x0, x1), y1)) → ACTIVE(and(active(splitAt(mark(x0), mark(x1))), y1))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(snd(X)) → MARK(X)
MARK(splitAt(cons(x0, x1), y1)) → ACTIVE(splitAt(active(cons(mark(x0), x1)), mark(y1)))
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(splitAt(s(x0), y1)) → ACTIVE(splitAt(active(s(mark(x0))), mark(y1)))
MARK(U12(splitAt(x0, x1), y1)) → ACTIVE(U12(active(splitAt(mark(x0), mark(x1))), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(splitAt(splitAt(x0, x1), y1)) → ACTIVE(splitAt(active(splitAt(mark(x0), mark(x1))), mark(y1)))
MARK(splitAt(afterNth(x0, x1), y1)) → ACTIVE(splitAt(active(afterNth(mark(x0), mark(x1))), mark(y1)))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), y1))
MARK(splitAt(head(x0), y1)) → ACTIVE(splitAt(active(head(mark(x0))), mark(y1)))
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))

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.
By narrowing [15] the rule MARK(tail(X)) → ACTIVE(tail(mark(X))) at position [0] we obtained the following new rules:

MARK(tail(fst(x0))) → ACTIVE(tail(active(fst(mark(x0)))))
MARK(tail(sel(x0, x1))) → ACTIVE(tail(active(sel(mark(x0), mark(x1)))))
MARK(tail(tail(x0))) → ACTIVE(tail(active(tail(mark(x0)))))
MARK(tail(pair(x0, x1))) → ACTIVE(tail(active(pair(mark(x0), mark(x1)))))
MARK(tail(U12(x0, x1))) → ACTIVE(tail(active(U12(mark(x0), x1))))
MARK(tail(take(x0, x1))) → ACTIVE(tail(active(take(mark(x0), mark(x1)))))
MARK(tail(afterNth(x0, x1))) → ACTIVE(tail(active(afterNth(mark(x0), mark(x1)))))
MARK(tail(nil)) → ACTIVE(tail(active(nil)))
MARK(tail(snd(x0))) → ACTIVE(tail(active(snd(mark(x0)))))
MARK(tail(natsFrom(x0))) → ACTIVE(tail(active(natsFrom(mark(x0)))))
MARK(tail(splitAt(x0, x1))) → ACTIVE(tail(active(splitAt(mark(x0), mark(x1)))))
MARK(tail(U11(x0, x1, x2, x3))) → ACTIVE(tail(active(U11(mark(x0), x1, x2, x3))))
MARK(tail(cons(x0, x1))) → ACTIVE(tail(active(cons(mark(x0), x1))))
MARK(tail(and(x0, x1))) → ACTIVE(tail(active(and(mark(x0), x1))))
MARK(tail(x0)) → ACTIVE(tail(x0))
MARK(tail(s(x0))) → ACTIVE(tail(active(s(mark(x0)))))
MARK(tail(0)) → ACTIVE(tail(active(0)))
MARK(tail(tt)) → ACTIVE(tail(active(tt)))
MARK(tail(head(x0))) → ACTIVE(tail(active(head(mark(x0)))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

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

MARK(splitAt(y0, U12(x0, x1))) → ACTIVE(splitAt(mark(y0), active(U12(mark(x0), x1))))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), y1))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), x1)), y1))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U11(tt, N, X, XS))
MARK(and(pair(x0, x1), y1)) → ACTIVE(and(active(pair(mark(x0), mark(x1))), y1))
ACTIVE(take(N, XS)) → MARK(fst(splitAt(N, XS)))
MARK(splitAt(natsFrom(x0), y1)) → ACTIVE(splitAt(active(natsFrom(mark(x0))), mark(y1)))
ACTIVE(sel(N, XS)) → MARK(head(afterNth(N, XS)))
MARK(and(nil, y1)) → ACTIVE(and(active(nil), y1))
MARK(tail(and(x0, x1))) → ACTIVE(tail(active(and(mark(x0), x1))))
MARK(fst(X)) → MARK(X)
MARK(splitAt(y0, take(x0, x1))) → ACTIVE(splitAt(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U12(pair(x0, x1), y1)) → ACTIVE(U12(active(pair(mark(x0), mark(x1))), y1))
MARK(tail(take(x0, x1))) → ACTIVE(tail(active(take(mark(x0), mark(x1)))))
MARK(splitAt(y0, splitAt(x0, x1))) → ACTIVE(splitAt(mark(y0), active(splitAt(mark(x0), mark(x1)))))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(splitAt(sel(x0, x1), y1)) → ACTIVE(splitAt(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(splitAt(y0, U11(x0, x1, x2, x3))) → ACTIVE(splitAt(mark(y0), active(U11(mark(x0), x1, x2, x3))))
MARK(and(afterNth(x0, x1), y1)) → ACTIVE(and(active(afterNth(mark(x0), mark(x1))), y1))
MARK(splitAt(y0, snd(x0))) → ACTIVE(splitAt(mark(y0), active(snd(mark(x0)))))
MARK(splitAt(y0, afterNth(x0, x1))) → ACTIVE(splitAt(mark(y0), active(afterNth(mark(x0), mark(x1)))))
MARK(and(x0, x1)) → ACTIVE(and(x0, x1))
MARK(tail(snd(x0))) → ACTIVE(tail(active(snd(mark(x0)))))
MARK(tail(nil)) → ACTIVE(tail(active(nil)))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(splitAt(y0, and(x0, x1))) → ACTIVE(splitAt(mark(y0), active(and(mark(x0), x1))))
ACTIVE(and(tt, X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
MARK(and(U11(x0, x1, x2, x3), y1)) → ACTIVE(and(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(tail(head(x0))) → ACTIVE(tail(active(head(mark(x0)))))
MARK(U12(U11(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(splitAt(y0, tt)) → ACTIVE(splitAt(mark(y0), active(tt)))
MARK(splitAt(tt, y1)) → ACTIVE(splitAt(active(tt), mark(y1)))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(U12(sel(x0, x1), y1)) → ACTIVE(U12(active(sel(mark(x0), mark(x1))), y1))
MARK(take(X1, X2)) → MARK(X2)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
MARK(U12(fst(x0), y1)) → ACTIVE(U12(active(fst(mark(x0))), y1))
MARK(s(X)) → MARK(X)
MARK(and(sel(x0, x1), y1)) → ACTIVE(and(active(sel(mark(x0), mark(x1))), y1))
MARK(and(snd(x0), y1)) → ACTIVE(and(active(snd(mark(x0))), y1))
MARK(splitAt(y0, pair(x0, x1))) → ACTIVE(splitAt(mark(y0), active(pair(mark(x0), mark(x1)))))
ACTIVE(U12(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(and(0, y1)) → ACTIVE(and(active(0), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(and(y0, mark(x1))) → ACTIVE(and(mark(y0), x1))
ACTIVE(U11(tt, N, X, XS)) → MARK(U12(splitAt(N, XS), X))
MARK(and(X1, X2)) → MARK(X1)
MARK(splitAt(y0, head(x0))) → ACTIVE(splitAt(mark(y0), active(head(mark(x0)))))
MARK(splitAt(y0, tail(x0))) → ACTIVE(splitAt(mark(y0), active(tail(mark(x0)))))
MARK(tail(cons(x0, x1))) → ACTIVE(tail(active(cons(mark(x0), x1))))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), y1))
MARK(splitAt(U11(x0, x1, x2, x3), y1)) → ACTIVE(splitAt(active(U11(mark(x0), x1, x2, x3)), mark(y1)))
MARK(splitAt(y0, nil)) → ACTIVE(splitAt(mark(y0), active(nil)))
MARK(splitAt(nil, y1)) → ACTIVE(splitAt(active(nil), mark(y1)))
MARK(tail(x0)) → ACTIVE(tail(x0))
MARK(tail(sel(x0, x1))) → ACTIVE(tail(active(sel(mark(x0), mark(x1)))))
MARK(tail(U12(x0, x1))) → ACTIVE(tail(active(U12(mark(x0), x1))))
MARK(splitAt(y0, x1)) → ACTIVE(splitAt(mark(y0), x1))
MARK(U12(natsFrom(x0), y1)) → ACTIVE(U12(active(natsFrom(mark(x0))), y1))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(splitAt(y0, cons(x0, x1))) → ACTIVE(splitAt(mark(y0), active(cons(mark(x0), x1))))
MARK(tail(U11(x0, x1, x2, x3))) → ACTIVE(tail(active(U11(mark(x0), x1, x2, x3))))
MARK(splitAt(snd(x0), y1)) → ACTIVE(splitAt(active(snd(mark(x0))), mark(y1)))
MARK(and(head(x0), y1)) → ACTIVE(and(active(head(mark(x0))), y1))
MARK(U12(tail(x0), y1)) → ACTIVE(U12(active(tail(mark(x0))), y1))
MARK(U11(X1, X2, X3, X4)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(splitAt(y0, 0)) → ACTIVE(splitAt(mark(y0), active(0)))
MARK(splitAt(0, y1)) → ACTIVE(splitAt(active(0), mark(y1)))
MARK(tail(tt)) → ACTIVE(tail(active(tt)))
ACTIVE(tail(cons(N, XS))) → MARK(XS)
MARK(tail(pair(x0, x1))) → ACTIVE(tail(active(pair(mark(x0), mark(x1)))))
MARK(head(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(U12(afterNth(x0, x1), y1)) → ACTIVE(U12(active(afterNth(mark(x0), mark(x1))), y1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(splitAt(and(x0, x1), y1)) → ACTIVE(splitAt(active(and(mark(x0), x1)), mark(y1)))
ACTIVE(afterNth(N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(and(U12(x0, x1), y1)) → ACTIVE(and(active(U12(mark(x0), x1)), y1))
MARK(splitAt(y0, natsFrom(x0))) → ACTIVE(splitAt(mark(y0), active(natsFrom(mark(x0)))))
MARK(take(X1, X2)) → MARK(X1)
MARK(U12(and(x0, x1), y1)) → ACTIVE(U12(active(and(mark(x0), x1)), y1))
ACTIVE(snd(pair(X, Y))) → MARK(Y)
MARK(splitAt(pair(x0, x1), y1)) → ACTIVE(splitAt(active(pair(mark(x0), mark(x1))), mark(y1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(splitAt(take(x0, x1), y1)) → ACTIVE(splitAt(active(take(mark(x0), mark(x1))), mark(y1)))
MARK(and(y0, active(x1))) → ACTIVE(and(mark(y0), x1))
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(splitAt(U12(x0, x1), y1)) → ACTIVE(splitAt(active(U12(mark(x0), x1)), mark(y1)))
MARK(U11(X1, X2, X3, X4)) → ACTIVE(U11(mark(X1), X2, X3, X4))
MARK(tail(afterNth(x0, x1))) → ACTIVE(tail(active(afterNth(mark(x0), mark(x1)))))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X1)
MARK(U12(snd(x0), y1)) → ACTIVE(U12(active(snd(mark(x0))), y1))
ACTIVE(fst(pair(X, Y))) → MARK(X)
MARK(splitAt(y0, fst(x0))) → ACTIVE(splitAt(mark(y0), active(fst(mark(x0)))))
MARK(tail(s(x0))) → ACTIVE(tail(active(s(mark(x0)))))
MARK(tail(0)) → ACTIVE(tail(active(0)))
MARK(and(tail(x0), y1)) → ACTIVE(and(active(tail(mark(x0))), y1))
MARK(afterNth(X1, X2)) → MARK(X1)
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(tail(fst(x0))) → ACTIVE(tail(active(fst(mark(x0)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(splitAt(x0, y1)) → ACTIVE(splitAt(x0, mark(y1)))
MARK(splitAt(fst(x0), y1)) → ACTIVE(splitAt(active(fst(mark(x0))), mark(y1)))
MARK(and(natsFrom(x0), y1)) → ACTIVE(and(active(natsFrom(mark(x0))), y1))
MARK(and(take(x0, x1), y1)) → ACTIVE(and(active(take(mark(x0), mark(x1))), y1))
MARK(U12(head(x0), y1)) → ACTIVE(U12(active(head(mark(x0))), y1))
MARK(natsFrom(X)) → MARK(X)
MARK(splitAt(y0, sel(x0, x1))) → ACTIVE(splitAt(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(tail(natsFrom(x0))) → ACTIVE(tail(active(natsFrom(mark(x0)))))
MARK(pair(X1, X2)) → MARK(X2)
MARK(and(fst(x0), y1)) → ACTIVE(and(active(fst(mark(x0))), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(splitAt(y0, s(x0))) → ACTIVE(splitAt(mark(y0), active(s(mark(x0)))))
MARK(and(splitAt(x0, x1), y1)) → ACTIVE(and(active(splitAt(mark(x0), mark(x1))), y1))
MARK(splitAt(tail(x0), y1)) → ACTIVE(splitAt(active(tail(mark(x0))), mark(y1)))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(snd(X)) → MARK(X)
MARK(tail(tail(x0))) → ACTIVE(tail(active(tail(mark(x0)))))
MARK(splitAt(cons(x0, x1), y1)) → ACTIVE(splitAt(active(cons(mark(x0), x1)), mark(y1)))
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(splitAt(s(x0), y1)) → ACTIVE(splitAt(active(s(mark(x0))), mark(y1)))
MARK(U12(splitAt(x0, x1), y1)) → ACTIVE(U12(active(splitAt(mark(x0), mark(x1))), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(splitAt(afterNth(x0, x1), y1)) → ACTIVE(splitAt(active(afterNth(mark(x0), mark(x1))), mark(y1)))
MARK(splitAt(splitAt(x0, x1), y1)) → ACTIVE(splitAt(active(splitAt(mark(x0), mark(x1))), mark(y1)))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), y1))
MARK(splitAt(head(x0), y1)) → ACTIVE(splitAt(active(head(mark(x0))), mark(y1)))
MARK(tail(splitAt(x0, x1))) → ACTIVE(tail(active(splitAt(mark(x0), mark(x1)))))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))

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.
By narrowing [15] the rule MARK(head(X)) → ACTIVE(head(mark(X))) at position [0] we obtained the following new rules:

MARK(head(fst(x0))) → ACTIVE(head(active(fst(mark(x0)))))
MARK(head(U12(x0, x1))) → ACTIVE(head(active(U12(mark(x0), x1))))
MARK(head(nil)) → ACTIVE(head(active(nil)))
MARK(head(sel(x0, x1))) → ACTIVE(head(active(sel(mark(x0), mark(x1)))))
MARK(head(tt)) → ACTIVE(head(active(tt)))
MARK(head(and(x0, x1))) → ACTIVE(head(active(and(mark(x0), x1))))
MARK(head(snd(x0))) → ACTIVE(head(active(snd(mark(x0)))))
MARK(head(splitAt(x0, x1))) → ACTIVE(head(active(splitAt(mark(x0), mark(x1)))))
MARK(head(x0)) → ACTIVE(head(x0))
MARK(head(s(x0))) → ACTIVE(head(active(s(mark(x0)))))
MARK(head(U11(x0, x1, x2, x3))) → ACTIVE(head(active(U11(mark(x0), x1, x2, x3))))
MARK(head(tail(x0))) → ACTIVE(head(active(tail(mark(x0)))))
MARK(head(0)) → ACTIVE(head(active(0)))
MARK(head(afterNth(x0, x1))) → ACTIVE(head(active(afterNth(mark(x0), mark(x1)))))
MARK(head(head(x0))) → ACTIVE(head(active(head(mark(x0)))))
MARK(head(pair(x0, x1))) → ACTIVE(head(active(pair(mark(x0), mark(x1)))))
MARK(head(natsFrom(x0))) → ACTIVE(head(active(natsFrom(mark(x0)))))
MARK(head(cons(x0, x1))) → ACTIVE(head(active(cons(mark(x0), x1))))
MARK(head(take(x0, x1))) → ACTIVE(head(active(take(mark(x0), mark(x1)))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

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

MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(splitAt(y0, U12(x0, x1))) → ACTIVE(splitAt(mark(y0), active(U12(mark(x0), x1))))
MARK(head(nil)) → ACTIVE(head(active(nil)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), x1)), y1))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), y1))
MARK(head(snd(x0))) → ACTIVE(head(active(snd(mark(x0)))))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U11(tt, N, X, XS))
MARK(and(pair(x0, x1), y1)) → ACTIVE(and(active(pair(mark(x0), mark(x1))), y1))
ACTIVE(take(N, XS)) → MARK(fst(splitAt(N, XS)))
MARK(head(0)) → ACTIVE(head(active(0)))
MARK(head(tail(x0))) → ACTIVE(head(active(tail(mark(x0)))))
ACTIVE(sel(N, XS)) → MARK(head(afterNth(N, XS)))
MARK(splitAt(natsFrom(x0), y1)) → ACTIVE(splitAt(active(natsFrom(mark(x0))), mark(y1)))
MARK(head(afterNth(x0, x1))) → ACTIVE(head(active(afterNth(mark(x0), mark(x1)))))
MARK(and(nil, y1)) → ACTIVE(and(active(nil), y1))
MARK(head(cons(x0, x1))) → ACTIVE(head(active(cons(mark(x0), x1))))
MARK(tail(and(x0, x1))) → ACTIVE(tail(active(and(mark(x0), x1))))
MARK(fst(X)) → MARK(X)
MARK(splitAt(y0, take(x0, x1))) → ACTIVE(splitAt(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U12(pair(x0, x1), y1)) → ACTIVE(U12(active(pair(mark(x0), mark(x1))), y1))
MARK(head(and(x0, x1))) → ACTIVE(head(active(and(mark(x0), x1))))
MARK(splitAt(y0, splitAt(x0, x1))) → ACTIVE(splitAt(mark(y0), active(splitAt(mark(x0), mark(x1)))))
MARK(tail(take(x0, x1))) → ACTIVE(tail(active(take(mark(x0), mark(x1)))))
MARK(splitAt(sel(x0, x1), y1)) → ACTIVE(splitAt(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(splitAt(y0, U11(x0, x1, x2, x3))) → ACTIVE(splitAt(mark(y0), active(U11(mark(x0), x1, x2, x3))))
MARK(head(x0)) → ACTIVE(head(x0))
MARK(splitAt(y0, snd(x0))) → ACTIVE(splitAt(mark(y0), active(snd(mark(x0)))))
MARK(and(afterNth(x0, x1), y1)) → ACTIVE(and(active(afterNth(mark(x0), mark(x1))), y1))
MARK(splitAt(y0, afterNth(x0, x1))) → ACTIVE(splitAt(mark(y0), active(afterNth(mark(x0), mark(x1)))))
MARK(head(U11(x0, x1, x2, x3))) → ACTIVE(head(active(U11(mark(x0), x1, x2, x3))))
MARK(and(x0, x1)) → ACTIVE(and(x0, x1))
MARK(tail(nil)) → ACTIVE(tail(active(nil)))
MARK(tail(snd(x0))) → ACTIVE(tail(active(snd(mark(x0)))))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(head(head(x0))) → ACTIVE(head(active(head(mark(x0)))))
MARK(splitAt(y0, and(x0, x1))) → ACTIVE(splitAt(mark(y0), active(and(mark(x0), x1))))
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
ACTIVE(and(tt, X)) → MARK(X)
MARK(and(U11(x0, x1, x2, x3), y1)) → ACTIVE(and(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(U12(U11(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(tail(head(x0))) → ACTIVE(tail(active(head(mark(x0)))))
MARK(U12(sel(x0, x1), y1)) → ACTIVE(U12(active(sel(mark(x0), mark(x1))), y1))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(splitAt(tt, y1)) → ACTIVE(splitAt(active(tt), mark(y1)))
MARK(splitAt(y0, tt)) → ACTIVE(splitAt(mark(y0), active(tt)))
MARK(take(X1, X2)) → MARK(X2)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
MARK(U12(fst(x0), y1)) → ACTIVE(U12(active(fst(mark(x0))), y1))
MARK(s(X)) → MARK(X)
MARK(head(tt)) → ACTIVE(head(active(tt)))
MARK(and(snd(x0), y1)) → ACTIVE(and(active(snd(mark(x0))), y1))
MARK(and(sel(x0, x1), y1)) → ACTIVE(and(active(sel(mark(x0), mark(x1))), y1))
MARK(splitAt(y0, pair(x0, x1))) → ACTIVE(splitAt(mark(y0), active(pair(mark(x0), mark(x1)))))
ACTIVE(U12(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(and(0, y1)) → ACTIVE(and(active(0), y1))
MARK(and(y0, mark(x1))) → ACTIVE(and(mark(y0), x1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
ACTIVE(U11(tt, N, X, XS)) → MARK(U12(splitAt(N, XS), X))
MARK(and(X1, X2)) → MARK(X1)
MARK(splitAt(y0, tail(x0))) → ACTIVE(splitAt(mark(y0), active(tail(mark(x0)))))
MARK(splitAt(y0, head(x0))) → ACTIVE(splitAt(mark(y0), active(head(mark(x0)))))
MARK(splitAt(U11(x0, x1, x2, x3), y1)) → ACTIVE(splitAt(active(U11(mark(x0), x1, x2, x3)), mark(y1)))
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(tail(cons(x0, x1))) → ACTIVE(tail(active(cons(mark(x0), x1))))
MARK(splitAt(nil, y1)) → ACTIVE(splitAt(active(nil), mark(y1)))
MARK(splitAt(y0, nil)) → ACTIVE(splitAt(mark(y0), active(nil)))
MARK(tail(x0)) → ACTIVE(tail(x0))
MARK(head(U12(x0, x1))) → ACTIVE(head(active(U12(mark(x0), x1))))
MARK(tail(sel(x0, x1))) → ACTIVE(tail(active(sel(mark(x0), mark(x1)))))
MARK(tail(U12(x0, x1))) → ACTIVE(tail(active(U12(mark(x0), x1))))
MARK(splitAt(y0, x1)) → ACTIVE(splitAt(mark(y0), x1))
MARK(U12(natsFrom(x0), y1)) → ACTIVE(U12(active(natsFrom(mark(x0))), y1))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(splitAt(y0, cons(x0, x1))) → ACTIVE(splitAt(mark(y0), active(cons(mark(x0), x1))))
MARK(splitAt(snd(x0), y1)) → ACTIVE(splitAt(active(snd(mark(x0))), mark(y1)))
MARK(tail(U11(x0, x1, x2, x3))) → ACTIVE(tail(active(U11(mark(x0), x1, x2, x3))))
MARK(head(natsFrom(x0))) → ACTIVE(head(active(natsFrom(mark(x0)))))
MARK(and(head(x0), y1)) → ACTIVE(and(active(head(mark(x0))), y1))
MARK(U12(tail(x0), y1)) → ACTIVE(U12(active(tail(mark(x0))), y1))
MARK(U11(X1, X2, X3, X4)) → MARK(X1)
MARK(splitAt(0, y1)) → ACTIVE(splitAt(active(0), mark(y1)))
MARK(splitAt(y0, 0)) → ACTIVE(splitAt(mark(y0), active(0)))
MARK(sel(X1, X2)) → MARK(X2)
ACTIVE(tail(cons(N, XS))) → MARK(XS)
MARK(tail(tt)) → ACTIVE(tail(active(tt)))
MARK(head(fst(x0))) → ACTIVE(head(active(fst(mark(x0)))))
MARK(tail(pair(x0, x1))) → ACTIVE(tail(active(pair(mark(x0), mark(x1)))))
MARK(tail(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U12(afterNth(x0, x1), y1)) → ACTIVE(U12(active(afterNth(mark(x0), mark(x1))), y1))
MARK(splitAt(and(x0, x1), y1)) → ACTIVE(splitAt(active(and(mark(x0), x1)), mark(y1)))
ACTIVE(afterNth(N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(and(U12(x0, x1), y1)) → ACTIVE(and(active(U12(mark(x0), x1)), y1))
MARK(splitAt(y0, natsFrom(x0))) → ACTIVE(splitAt(mark(y0), active(natsFrom(mark(x0)))))
MARK(take(X1, X2)) → MARK(X1)
ACTIVE(snd(pair(X, Y))) → MARK(Y)
MARK(U12(and(x0, x1), y1)) → ACTIVE(U12(active(and(mark(x0), x1)), y1))
MARK(splitAt(pair(x0, x1), y1)) → ACTIVE(splitAt(active(pair(mark(x0), mark(x1))), mark(y1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(splitAt(take(x0, x1), y1)) → ACTIVE(splitAt(active(take(mark(x0), mark(x1))), mark(y1)))
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(and(y0, active(x1))) → ACTIVE(and(mark(y0), x1))
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(splitAt(U12(x0, x1), y1)) → ACTIVE(splitAt(active(U12(mark(x0), x1)), mark(y1)))
MARK(U11(X1, X2, X3, X4)) → ACTIVE(U11(mark(X1), X2, X3, X4))
MARK(pair(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(tail(afterNth(x0, x1))) → ACTIVE(tail(active(afterNth(mark(x0), mark(x1)))))
MARK(head(splitAt(x0, x1))) → ACTIVE(head(active(splitAt(mark(x0), mark(x1)))))
MARK(U12(snd(x0), y1)) → ACTIVE(U12(active(snd(mark(x0))), y1))
ACTIVE(fst(pair(X, Y))) → MARK(X)
MARK(head(pair(x0, x1))) → ACTIVE(head(active(pair(mark(x0), mark(x1)))))
MARK(splitAt(y0, fst(x0))) → ACTIVE(splitAt(mark(y0), active(fst(mark(x0)))))
MARK(tail(s(x0))) → ACTIVE(tail(active(s(mark(x0)))))
MARK(and(tail(x0), y1)) → ACTIVE(and(active(tail(mark(x0))), y1))
MARK(tail(0)) → ACTIVE(tail(active(0)))
MARK(afterNth(X1, X2)) → MARK(X1)
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(tail(fst(x0))) → ACTIVE(tail(active(fst(mark(x0)))))
MARK(splitAt(x0, y1)) → ACTIVE(splitAt(x0, mark(y1)))
MARK(and(natsFrom(x0), y1)) → ACTIVE(and(active(natsFrom(mark(x0))), y1))
MARK(splitAt(fst(x0), y1)) → ACTIVE(splitAt(active(fst(mark(x0))), mark(y1)))
MARK(and(take(x0, x1), y1)) → ACTIVE(and(active(take(mark(x0), mark(x1))), y1))
MARK(U12(head(x0), y1)) → ACTIVE(U12(active(head(mark(x0))), y1))
MARK(natsFrom(X)) → MARK(X)
MARK(splitAt(y0, sel(x0, x1))) → ACTIVE(splitAt(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(tail(natsFrom(x0))) → ACTIVE(tail(active(natsFrom(mark(x0)))))
MARK(pair(X1, X2)) → MARK(X2)
MARK(and(fst(x0), y1)) → ACTIVE(and(active(fst(mark(x0))), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(splitAt(y0, s(x0))) → ACTIVE(splitAt(mark(y0), active(s(mark(x0)))))
MARK(splitAt(tail(x0), y1)) → ACTIVE(splitAt(active(tail(mark(x0))), mark(y1)))
MARK(and(splitAt(x0, x1), y1)) → ACTIVE(and(active(splitAt(mark(x0), mark(x1))), y1))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(snd(X)) → MARK(X)
MARK(head(sel(x0, x1))) → ACTIVE(head(active(sel(mark(x0), mark(x1)))))
MARK(tail(tail(x0))) → ACTIVE(tail(active(tail(mark(x0)))))
MARK(splitAt(cons(x0, x1), y1)) → ACTIVE(splitAt(active(cons(mark(x0), x1)), mark(y1)))
MARK(splitAt(s(x0), y1)) → ACTIVE(splitAt(active(s(mark(x0))), mark(y1)))
MARK(U12(splitAt(x0, x1), y1)) → ACTIVE(U12(active(splitAt(mark(x0), mark(x1))), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(head(s(x0))) → ACTIVE(head(active(s(mark(x0)))))
MARK(splitAt(splitAt(x0, x1), y1)) → ACTIVE(splitAt(active(splitAt(mark(x0), mark(x1))), mark(y1)))
MARK(splitAt(afterNth(x0, x1), y1)) → ACTIVE(splitAt(active(afterNth(mark(x0), mark(x1))), mark(y1)))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), y1))
MARK(splitAt(head(x0), y1)) → ACTIVE(splitAt(active(head(mark(x0))), mark(y1)))
MARK(tail(splitAt(x0, x1))) → ACTIVE(tail(active(splitAt(mark(x0), mark(x1)))))
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(head(take(x0, x1))) → ACTIVE(head(active(take(mark(x0), mark(x1)))))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))

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.
By narrowing [15] the rule MARK(U11(X1, X2, X3, X4)) → ACTIVE(U11(mark(X1), X2, X3, X4)) at position [0] we obtained the following new rules:

MARK(U11(natsFrom(x0), y1, y2, y3)) → ACTIVE(U11(active(natsFrom(mark(x0))), y1, y2, y3))
MARK(U11(y0, x1, x2, mark(x3))) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U11(x0, x1, x2, x3)) → ACTIVE(U11(x0, x1, x2, x3))
MARK(U11(sel(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(sel(mark(x0), mark(x1))), y1, y2, y3))
MARK(U11(0, y1, y2, y3)) → ACTIVE(U11(active(0), y1, y2, y3))
MARK(U11(cons(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1, y2, y3))
MARK(U11(U11(x0, x1, x2, x3), y1, y2, y3)) → ACTIVE(U11(active(U11(mark(x0), x1, x2, x3)), y1, y2, y3))
MARK(U11(and(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(and(mark(x0), x1)), y1, y2, y3))
MARK(U11(y0, mark(x1), x2, x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U11(fst(x0), y1, y2, y3)) → ACTIVE(U11(active(fst(mark(x0))), y1, y2, y3))
MARK(U11(pair(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(pair(mark(x0), mark(x1))), y1, y2, y3))
MARK(U11(U12(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1, y2, y3))
MARK(U11(splitAt(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(splitAt(mark(x0), mark(x1))), y1, y2, y3))
MARK(U11(y0, x1, mark(x2), x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U11(snd(x0), y1, y2, y3)) → ACTIVE(U11(active(snd(mark(x0))), y1, y2, y3))
MARK(U11(y0, x1, x2, active(x3))) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U11(y0, x1, active(x2), x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U11(s(x0), y1, y2, y3)) → ACTIVE(U11(active(s(mark(x0))), y1, y2, y3))
MARK(U11(tt, y1, y2, y3)) → ACTIVE(U11(active(tt), y1, y2, y3))
MARK(U11(tail(x0), y1, y2, y3)) → ACTIVE(U11(active(tail(mark(x0))), y1, y2, y3))
MARK(U11(y0, active(x1), x2, x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U11(take(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1, y2, y3))
MARK(U11(head(x0), y1, y2, y3)) → ACTIVE(U11(active(head(mark(x0))), y1, y2, y3))
MARK(U11(nil, y1, y2, y3)) → ACTIVE(U11(active(nil), y1, y2, y3))
MARK(U11(afterNth(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(afterNth(mark(x0), mark(x1))), y1, y2, y3))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

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

MARK(U11(natsFrom(x0), y1, y2, y3)) → ACTIVE(U11(active(natsFrom(mark(x0))), y1, y2, y3))
MARK(splitAt(y0, U12(x0, x1))) → ACTIVE(splitAt(mark(y0), active(U12(mark(x0), x1))))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(head(nil)) → ACTIVE(head(active(nil)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), y1))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), x1)), y1))
MARK(head(snd(x0))) → ACTIVE(head(active(snd(mark(x0)))))
MARK(U11(and(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(and(mark(x0), x1)), y1, y2, y3))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U11(tt, N, X, XS))
MARK(U11(y0, mark(x1), x2, x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(and(pair(x0, x1), y1)) → ACTIVE(and(active(pair(mark(x0), mark(x1))), y1))
ACTIVE(take(N, XS)) → MARK(fst(splitAt(N, XS)))
MARK(splitAt(natsFrom(x0), y1)) → ACTIVE(splitAt(active(natsFrom(mark(x0))), mark(y1)))
ACTIVE(sel(N, XS)) → MARK(head(afterNth(N, XS)))
MARK(head(tail(x0))) → ACTIVE(head(active(tail(mark(x0)))))
MARK(head(0)) → ACTIVE(head(active(0)))
MARK(head(afterNth(x0, x1))) → ACTIVE(head(active(afterNth(mark(x0), mark(x1)))))
MARK(and(nil, y1)) → ACTIVE(and(active(nil), y1))
MARK(tail(and(x0, x1))) → ACTIVE(tail(active(and(mark(x0), x1))))
MARK(head(cons(x0, x1))) → ACTIVE(head(active(cons(mark(x0), x1))))
MARK(fst(X)) → MARK(X)
MARK(splitAt(y0, take(x0, x1))) → ACTIVE(splitAt(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(splitAt(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(splitAt(mark(x0), mark(x1))), y1, y2, y3))
MARK(U12(pair(x0, x1), y1)) → ACTIVE(U12(active(pair(mark(x0), mark(x1))), y1))
MARK(tail(take(x0, x1))) → ACTIVE(tail(active(take(mark(x0), mark(x1)))))
MARK(splitAt(y0, splitAt(x0, x1))) → ACTIVE(splitAt(mark(y0), active(splitAt(mark(x0), mark(x1)))))
MARK(head(and(x0, x1))) → ACTIVE(head(active(and(mark(x0), x1))))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(splitAt(sel(x0, x1), y1)) → ACTIVE(splitAt(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(U11(y0, x1, active(x2), x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(splitAt(y0, U11(x0, x1, x2, x3))) → ACTIVE(splitAt(mark(y0), active(U11(mark(x0), x1, x2, x3))))
MARK(and(afterNth(x0, x1), y1)) → ACTIVE(and(active(afterNth(mark(x0), mark(x1))), y1))
MARK(splitAt(y0, snd(x0))) → ACTIVE(splitAt(mark(y0), active(snd(mark(x0)))))
MARK(head(x0)) → ACTIVE(head(x0))
MARK(splitAt(y0, afterNth(x0, x1))) → ACTIVE(splitAt(mark(y0), active(afterNth(mark(x0), mark(x1)))))
MARK(and(x0, x1)) → ACTIVE(and(x0, x1))
MARK(head(U11(x0, x1, x2, x3))) → ACTIVE(head(active(U11(mark(x0), x1, x2, x3))))
MARK(tail(snd(x0))) → ACTIVE(tail(active(snd(mark(x0)))))
MARK(tail(nil)) → ACTIVE(tail(active(nil)))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(head(head(x0))) → ACTIVE(head(active(head(mark(x0)))))
MARK(U11(head(x0), y1, y2, y3)) → ACTIVE(U11(active(head(mark(x0))), y1, y2, y3))
MARK(splitAt(y0, and(x0, x1))) → ACTIVE(splitAt(mark(y0), active(and(mark(x0), x1))))
MARK(U11(afterNth(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(afterNth(mark(x0), mark(x1))), y1, y2, y3))
ACTIVE(and(tt, X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
MARK(and(U11(x0, x1, x2, x3), y1)) → ACTIVE(and(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(tail(head(x0))) → ACTIVE(tail(active(head(mark(x0)))))
MARK(U12(U11(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(splitAt(y0, tt)) → ACTIVE(splitAt(mark(y0), active(tt)))
MARK(splitAt(tt, y1)) → ACTIVE(splitAt(active(tt), mark(y1)))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(U12(sel(x0, x1), y1)) → ACTIVE(U12(active(sel(mark(x0), mark(x1))), y1))
MARK(take(X1, X2)) → MARK(X2)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
MARK(U12(fst(x0), y1)) → ACTIVE(U12(active(fst(mark(x0))), y1))
MARK(s(X)) → MARK(X)
MARK(and(sel(x0, x1), y1)) → ACTIVE(and(active(sel(mark(x0), mark(x1))), y1))
MARK(and(snd(x0), y1)) → ACTIVE(and(active(snd(mark(x0))), y1))
MARK(head(tt)) → ACTIVE(head(active(tt)))
MARK(U11(cons(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1, y2, y3))
MARK(splitAt(y0, pair(x0, x1))) → ACTIVE(splitAt(mark(y0), active(pair(mark(x0), mark(x1)))))
ACTIVE(U12(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(and(0, y1)) → ACTIVE(and(active(0), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(and(y0, mark(x1))) → ACTIVE(and(mark(y0), x1))
ACTIVE(U11(tt, N, X, XS)) → MARK(U12(splitAt(N, XS), X))
MARK(and(X1, X2)) → MARK(X1)
MARK(splitAt(y0, head(x0))) → ACTIVE(splitAt(mark(y0), active(head(mark(x0)))))
MARK(splitAt(y0, tail(x0))) → ACTIVE(splitAt(mark(y0), active(tail(mark(x0)))))
MARK(tail(cons(x0, x1))) → ACTIVE(tail(active(cons(mark(x0), x1))))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), y1))
MARK(splitAt(U11(x0, x1, x2, x3), y1)) → ACTIVE(splitAt(active(U11(mark(x0), x1, x2, x3)), mark(y1)))
MARK(splitAt(y0, nil)) → ACTIVE(splitAt(mark(y0), active(nil)))
MARK(splitAt(nil, y1)) → ACTIVE(splitAt(active(nil), mark(y1)))
MARK(U11(pair(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(pair(mark(x0), mark(x1))), y1, y2, y3))
MARK(tail(x0)) → ACTIVE(tail(x0))
MARK(tail(sel(x0, x1))) → ACTIVE(tail(active(sel(mark(x0), mark(x1)))))
MARK(head(U12(x0, x1))) → ACTIVE(head(active(U12(mark(x0), x1))))
MARK(tail(U12(x0, x1))) → ACTIVE(tail(active(U12(mark(x0), x1))))
MARK(splitAt(y0, x1)) → ACTIVE(splitAt(mark(y0), x1))
MARK(U12(natsFrom(x0), y1)) → ACTIVE(U12(active(natsFrom(mark(x0))), y1))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(U11(tt, y1, y2, y3)) → ACTIVE(U11(active(tt), y1, y2, y3))
MARK(U11(take(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1, y2, y3))
MARK(splitAt(y0, cons(x0, x1))) → ACTIVE(splitAt(mark(y0), active(cons(mark(x0), x1))))
MARK(tail(U11(x0, x1, x2, x3))) → ACTIVE(tail(active(U11(mark(x0), x1, x2, x3))))
MARK(splitAt(snd(x0), y1)) → ACTIVE(splitAt(active(snd(mark(x0))), mark(y1)))
MARK(head(natsFrom(x0))) → ACTIVE(head(active(natsFrom(mark(x0)))))
MARK(and(head(x0), y1)) → ACTIVE(and(active(head(mark(x0))), y1))
MARK(U12(tail(x0), y1)) → ACTIVE(U12(active(tail(mark(x0))), y1))
MARK(U11(X1, X2, X3, X4)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(splitAt(y0, 0)) → ACTIVE(splitAt(mark(y0), active(0)))
MARK(splitAt(0, y1)) → ACTIVE(splitAt(active(0), mark(y1)))
MARK(tail(tt)) → ACTIVE(tail(active(tt)))
ACTIVE(tail(cons(N, XS))) → MARK(XS)
MARK(head(fst(x0))) → ACTIVE(head(active(fst(mark(x0)))))
MARK(U11(y0, x1, x2, mark(x3))) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U11(x0, x1, x2, x3)) → ACTIVE(U11(x0, x1, x2, x3))
MARK(U11(0, y1, y2, y3)) → ACTIVE(U11(active(0), y1, y2, y3))
MARK(tail(pair(x0, x1))) → ACTIVE(tail(active(pair(mark(x0), mark(x1)))))
MARK(head(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(U12(afterNth(x0, x1), y1)) → ACTIVE(U12(active(afterNth(mark(x0), mark(x1))), y1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U11(U11(x0, x1, x2, x3), y1, y2, y3)) → ACTIVE(U11(active(U11(mark(x0), x1, x2, x3)), y1, y2, y3))
MARK(splitAt(and(x0, x1), y1)) → ACTIVE(splitAt(active(and(mark(x0), x1)), mark(y1)))
ACTIVE(afterNth(N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(and(U12(x0, x1), y1)) → ACTIVE(and(active(U12(mark(x0), x1)), y1))
MARK(splitAt(y0, natsFrom(x0))) → ACTIVE(splitAt(mark(y0), active(natsFrom(mark(x0)))))
MARK(take(X1, X2)) → MARK(X1)
MARK(U12(and(x0, x1), y1)) → ACTIVE(U12(active(and(mark(x0), x1)), y1))
ACTIVE(snd(pair(X, Y))) → MARK(Y)
MARK(splitAt(pair(x0, x1), y1)) → ACTIVE(splitAt(active(pair(mark(x0), mark(x1))), mark(y1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(splitAt(take(x0, x1), y1)) → ACTIVE(splitAt(active(take(mark(x0), mark(x1))), mark(y1)))
MARK(and(y0, active(x1))) → ACTIVE(and(mark(y0), x1))
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(splitAt(U12(x0, x1), y1)) → ACTIVE(splitAt(active(U12(mark(x0), x1)), mark(y1)))
MARK(tail(afterNth(x0, x1))) → ACTIVE(tail(active(afterNth(mark(x0), mark(x1)))))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X1)
MARK(head(splitAt(x0, x1))) → ACTIVE(head(active(splitAt(mark(x0), mark(x1)))))
MARK(U12(snd(x0), y1)) → ACTIVE(U12(active(snd(mark(x0))), y1))
MARK(U11(y0, active(x1), x2, x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
ACTIVE(fst(pair(X, Y))) → MARK(X)
MARK(head(pair(x0, x1))) → ACTIVE(head(active(pair(mark(x0), mark(x1)))))
MARK(splitAt(y0, fst(x0))) → ACTIVE(splitAt(mark(y0), active(fst(mark(x0)))))
MARK(tail(s(x0))) → ACTIVE(tail(active(s(mark(x0)))))
MARK(tail(0)) → ACTIVE(tail(active(0)))
MARK(and(tail(x0), y1)) → ACTIVE(and(active(tail(mark(x0))), y1))
MARK(afterNth(X1, X2)) → MARK(X1)
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(tail(fst(x0))) → ACTIVE(tail(active(fst(mark(x0)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(splitAt(x0, y1)) → ACTIVE(splitAt(x0, mark(y1)))
MARK(U11(sel(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(sel(mark(x0), mark(x1))), y1, y2, y3))
MARK(splitAt(fst(x0), y1)) → ACTIVE(splitAt(active(fst(mark(x0))), mark(y1)))
MARK(and(natsFrom(x0), y1)) → ACTIVE(and(active(natsFrom(mark(x0))), y1))
MARK(and(take(x0, x1), y1)) → ACTIVE(and(active(take(mark(x0), mark(x1))), y1))
MARK(U12(head(x0), y1)) → ACTIVE(U12(active(head(mark(x0))), y1))
MARK(natsFrom(X)) → MARK(X)
MARK(splitAt(y0, sel(x0, x1))) → ACTIVE(splitAt(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(tail(natsFrom(x0))) → ACTIVE(tail(active(natsFrom(mark(x0)))))
MARK(U11(fst(x0), y1, y2, y3)) → ACTIVE(U11(active(fst(mark(x0))), y1, y2, y3))
MARK(pair(X1, X2)) → MARK(X2)
MARK(and(fst(x0), y1)) → ACTIVE(and(active(fst(mark(x0))), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(splitAt(y0, s(x0))) → ACTIVE(splitAt(mark(y0), active(s(mark(x0)))))
MARK(and(splitAt(x0, x1), y1)) → ACTIVE(and(active(splitAt(mark(x0), mark(x1))), y1))
MARK(splitAt(tail(x0), y1)) → ACTIVE(splitAt(active(tail(mark(x0))), mark(y1)))
MARK(U11(U12(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1, y2, y3))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(snd(X)) → MARK(X)
MARK(head(sel(x0, x1))) → ACTIVE(head(active(sel(mark(x0), mark(x1)))))
MARK(tail(tail(x0))) → ACTIVE(tail(active(tail(mark(x0)))))
MARK(splitAt(cons(x0, x1), y1)) → ACTIVE(splitAt(active(cons(mark(x0), x1)), mark(y1)))
MARK(splitAt(s(x0), y1)) → ACTIVE(splitAt(active(s(mark(x0))), mark(y1)))
MARK(U11(snd(x0), y1, y2, y3)) → ACTIVE(U11(active(snd(mark(x0))), y1, y2, y3))
MARK(U11(y0, x1, mark(x2), x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U11(y0, x1, x2, active(x3))) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U11(s(x0), y1, y2, y3)) → ACTIVE(U11(active(s(mark(x0))), y1, y2, y3))
MARK(U12(splitAt(x0, x1), y1)) → ACTIVE(U12(active(splitAt(mark(x0), mark(x1))), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(tail(x0), y1, y2, y3)) → ACTIVE(U11(active(tail(mark(x0))), y1, y2, y3))
MARK(splitAt(afterNth(x0, x1), y1)) → ACTIVE(splitAt(active(afterNth(mark(x0), mark(x1))), mark(y1)))
MARK(splitAt(splitAt(x0, x1), y1)) → ACTIVE(splitAt(active(splitAt(mark(x0), mark(x1))), mark(y1)))
MARK(head(s(x0))) → ACTIVE(head(active(s(mark(x0)))))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), y1))
MARK(splitAt(head(x0), y1)) → ACTIVE(splitAt(active(head(mark(x0))), mark(y1)))
MARK(tail(splitAt(x0, x1))) → ACTIVE(tail(active(splitAt(mark(x0), mark(x1)))))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(U11(nil, y1, y2, y3)) → ACTIVE(U11(active(nil), y1, y2, y3))
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(head(take(x0, x1))) → ACTIVE(head(active(take(mark(x0), mark(x1)))))

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.
By narrowing [15] the rule MARK(fst(X)) → ACTIVE(fst(mark(X))) at position [0] we obtained the following new rules:

MARK(fst(snd(x0))) → ACTIVE(fst(active(snd(mark(x0)))))
MARK(fst(tt)) → ACTIVE(fst(active(tt)))
MARK(fst(tail(x0))) → ACTIVE(fst(active(tail(mark(x0)))))
MARK(fst(take(x0, x1))) → ACTIVE(fst(active(take(mark(x0), mark(x1)))))
MARK(fst(x0)) → ACTIVE(fst(x0))
MARK(fst(U12(x0, x1))) → ACTIVE(fst(active(U12(mark(x0), x1))))
MARK(fst(and(x0, x1))) → ACTIVE(fst(active(and(mark(x0), x1))))
MARK(fst(s(x0))) → ACTIVE(fst(active(s(mark(x0)))))
MARK(fst(sel(x0, x1))) → ACTIVE(fst(active(sel(mark(x0), mark(x1)))))
MARK(fst(cons(x0, x1))) → ACTIVE(fst(active(cons(mark(x0), x1))))
MARK(fst(afterNth(x0, x1))) → ACTIVE(fst(active(afterNth(mark(x0), mark(x1)))))
MARK(fst(fst(x0))) → ACTIVE(fst(active(fst(mark(x0)))))
MARK(fst(nil)) → ACTIVE(fst(active(nil)))
MARK(fst(0)) → ACTIVE(fst(active(0)))
MARK(fst(pair(x0, x1))) → ACTIVE(fst(active(pair(mark(x0), mark(x1)))))
MARK(fst(U11(x0, x1, x2, x3))) → ACTIVE(fst(active(U11(mark(x0), x1, x2, x3))))
MARK(fst(natsFrom(x0))) → ACTIVE(fst(active(natsFrom(mark(x0)))))
MARK(fst(splitAt(x0, x1))) → ACTIVE(fst(active(splitAt(mark(x0), mark(x1)))))
MARK(fst(head(x0))) → ACTIVE(fst(active(head(mark(x0)))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

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

MARK(U11(natsFrom(x0), y1, y2, y3)) → ACTIVE(U11(active(natsFrom(mark(x0))), y1, y2, y3))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), y1))
MARK(head(snd(x0))) → ACTIVE(head(active(snd(mark(x0)))))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U11(tt, N, X, XS))
MARK(U11(y0, mark(x1), x2, x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(and(pair(x0, x1), y1)) → ACTIVE(and(active(pair(mark(x0), mark(x1))), y1))
ACTIVE(take(N, XS)) → MARK(fst(splitAt(N, XS)))
ACTIVE(sel(N, XS)) → MARK(head(afterNth(N, XS)))
MARK(and(nil, y1)) → ACTIVE(and(active(nil), y1))
MARK(fst(head(x0))) → ACTIVE(fst(active(head(mark(x0)))))
MARK(splitAt(y0, take(x0, x1))) → ACTIVE(splitAt(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(U11(splitAt(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(splitAt(mark(x0), mark(x1))), y1, y2, y3))
MARK(U12(pair(x0, x1), y1)) → ACTIVE(U12(active(pair(mark(x0), mark(x1))), y1))
MARK(U11(y0, x1, active(x2), x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(splitAt(y0, U11(x0, x1, x2, x3))) → ACTIVE(splitAt(mark(y0), active(U11(mark(x0), x1, x2, x3))))
MARK(splitAt(y0, snd(x0))) → ACTIVE(splitAt(mark(y0), active(snd(mark(x0)))))
MARK(head(x0)) → ACTIVE(head(x0))
MARK(splitAt(y0, afterNth(x0, x1))) → ACTIVE(splitAt(mark(y0), active(afterNth(mark(x0), mark(x1)))))
MARK(fst(cons(x0, x1))) → ACTIVE(fst(active(cons(mark(x0), x1))))
MARK(head(U11(x0, x1, x2, x3))) → ACTIVE(head(active(U11(mark(x0), x1, x2, x3))))
MARK(tail(nil)) → ACTIVE(tail(active(nil)))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(head(head(x0))) → ACTIVE(head(active(head(mark(x0)))))
MARK(splitAt(y0, and(x0, x1))) → ACTIVE(splitAt(mark(y0), active(and(mark(x0), x1))))
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
MARK(tail(head(x0))) → ACTIVE(tail(active(head(mark(x0)))))
MARK(U12(U11(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(U12(sel(x0, x1), y1)) → ACTIVE(U12(active(sel(mark(x0), mark(x1))), y1))
MARK(take(X1, X2)) → MARK(X2)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
MARK(and(snd(x0), y1)) → ACTIVE(and(active(snd(mark(x0))), y1))
MARK(splitAt(y0, pair(x0, x1))) → ACTIVE(splitAt(mark(y0), active(pair(mark(x0), mark(x1)))))
ACTIVE(U12(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(fst(0)) → ACTIVE(fst(active(0)))
MARK(tail(cons(x0, x1))) → ACTIVE(tail(active(cons(mark(x0), x1))))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), y1))
MARK(U11(pair(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(pair(mark(x0), mark(x1))), y1, y2, y3))
MARK(tail(x0)) → ACTIVE(tail(x0))
MARK(fst(snd(x0))) → ACTIVE(fst(active(snd(mark(x0)))))
MARK(tail(sel(x0, x1))) → ACTIVE(tail(active(sel(mark(x0), mark(x1)))))
MARK(head(U12(x0, x1))) → ACTIVE(head(active(U12(mark(x0), x1))))
MARK(tail(U12(x0, x1))) → ACTIVE(tail(active(U12(mark(x0), x1))))
MARK(fst(s(x0))) → ACTIVE(fst(active(s(mark(x0)))))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(U11(tt, y1, y2, y3)) → ACTIVE(U11(active(tt), y1, y2, y3))
MARK(tail(U11(x0, x1, x2, x3))) → ACTIVE(tail(active(U11(mark(x0), x1, x2, x3))))
MARK(head(natsFrom(x0))) → ACTIVE(head(active(natsFrom(mark(x0)))))
MARK(splitAt(y0, 0)) → ACTIVE(splitAt(mark(y0), active(0)))
MARK(splitAt(0, y1)) → ACTIVE(splitAt(active(0), mark(y1)))
MARK(tail(tt)) → ACTIVE(tail(active(tt)))
MARK(head(fst(x0))) → ACTIVE(head(active(fst(mark(x0)))))
MARK(U11(x0, x1, x2, x3)) → ACTIVE(U11(x0, x1, x2, x3))
MARK(U12(afterNth(x0, x1), y1)) → ACTIVE(U12(active(afterNth(mark(x0), mark(x1))), y1))
MARK(and(U12(x0, x1), y1)) → ACTIVE(and(active(U12(mark(x0), x1)), y1))
MARK(splitAt(y0, natsFrom(x0))) → ACTIVE(splitAt(mark(y0), active(natsFrom(mark(x0)))))
MARK(take(X1, X2)) → MARK(X1)
MARK(U12(and(x0, x1), y1)) → ACTIVE(U12(active(and(mark(x0), x1)), y1))
MARK(fst(natsFrom(x0))) → ACTIVE(fst(active(natsFrom(mark(x0)))))
MARK(fst(splitAt(x0, x1))) → ACTIVE(fst(active(splitAt(mark(x0), mark(x1)))))
MARK(sel(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(head(splitAt(x0, x1))) → ACTIVE(head(active(splitAt(mark(x0), mark(x1)))))
MARK(U11(y0, active(x1), x2, x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(fst(afterNth(x0, x1))) → ACTIVE(fst(active(afterNth(mark(x0), mark(x1)))))
ACTIVE(fst(pair(X, Y))) → MARK(X)
MARK(head(pair(x0, x1))) → ACTIVE(head(active(pair(mark(x0), mark(x1)))))
MARK(splitAt(y0, fst(x0))) → ACTIVE(splitAt(mark(y0), active(fst(mark(x0)))))
MARK(tail(s(x0))) → ACTIVE(tail(active(s(mark(x0)))))
MARK(tail(0)) → ACTIVE(tail(active(0)))
MARK(afterNth(X1, X2)) → MARK(X1)
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(tail(fst(x0))) → ACTIVE(tail(active(fst(mark(x0)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(splitAt(x0, y1)) → ACTIVE(splitAt(x0, mark(y1)))
MARK(fst(tail(x0))) → ACTIVE(fst(active(tail(mark(x0)))))
MARK(U11(sel(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(sel(mark(x0), mark(x1))), y1, y2, y3))
MARK(splitAt(fst(x0), y1)) → ACTIVE(splitAt(active(fst(mark(x0))), mark(y1)))
MARK(and(natsFrom(x0), y1)) → ACTIVE(and(active(natsFrom(mark(x0))), y1))
MARK(U12(head(x0), y1)) → ACTIVE(U12(active(head(mark(x0))), y1))
MARK(natsFrom(X)) → MARK(X)
MARK(splitAt(y0, sel(x0, x1))) → ACTIVE(splitAt(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(tail(natsFrom(x0))) → ACTIVE(tail(active(natsFrom(mark(x0)))))
MARK(and(fst(x0), y1)) → ACTIVE(and(active(fst(mark(x0))), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(splitAt(tail(x0), y1)) → ACTIVE(splitAt(active(tail(mark(x0))), mark(y1)))
MARK(U11(U12(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1, y2, y3))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(tail(tail(x0))) → ACTIVE(tail(active(tail(mark(x0)))))
MARK(U11(y0, x1, mark(x2), x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U11(y0, x1, x2, active(x3))) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(tail(x0), y1, y2, y3)) → ACTIVE(U11(active(tail(mark(x0))), y1, y2, y3))
MARK(splitAt(splitAt(x0, x1), y1)) → ACTIVE(splitAt(active(splitAt(mark(x0), mark(x1))), mark(y1)))
MARK(head(s(x0))) → ACTIVE(head(active(s(mark(x0)))))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), y1))
MARK(splitAt(head(x0), y1)) → ACTIVE(splitAt(active(head(mark(x0))), mark(y1)))
MARK(tail(splitAt(x0, x1))) → ACTIVE(tail(active(splitAt(mark(x0), mark(x1)))))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(head(take(x0, x1))) → ACTIVE(head(active(take(mark(x0), mark(x1)))))
MARK(splitAt(y0, U12(x0, x1))) → ACTIVE(splitAt(mark(y0), active(U12(mark(x0), x1))))
MARK(head(nil)) → ACTIVE(head(active(nil)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), x1)), y1))
MARK(U11(and(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(and(mark(x0), x1)), y1, y2, y3))
MARK(splitAt(natsFrom(x0), y1)) → ACTIVE(splitAt(active(natsFrom(mark(x0))), mark(y1)))
MARK(head(tail(x0))) → ACTIVE(head(active(tail(mark(x0)))))
MARK(head(0)) → ACTIVE(head(active(0)))
MARK(head(afterNth(x0, x1))) → ACTIVE(head(active(afterNth(mark(x0), mark(x1)))))
MARK(tail(and(x0, x1))) → ACTIVE(tail(active(and(mark(x0), x1))))
MARK(head(cons(x0, x1))) → ACTIVE(head(active(cons(mark(x0), x1))))
MARK(fst(X)) → MARK(X)
MARK(U12(X1, X2)) → MARK(X1)
MARK(fst(tt)) → ACTIVE(fst(active(tt)))
MARK(tail(take(x0, x1))) → ACTIVE(tail(active(take(mark(x0), mark(x1)))))
MARK(splitAt(y0, splitAt(x0, x1))) → ACTIVE(splitAt(mark(y0), active(splitAt(mark(x0), mark(x1)))))
MARK(head(and(x0, x1))) → ACTIVE(head(active(and(mark(x0), x1))))
MARK(fst(and(x0, x1))) → ACTIVE(fst(active(and(mark(x0), x1))))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(splitAt(sel(x0, x1), y1)) → ACTIVE(splitAt(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(and(afterNth(x0, x1), y1)) → ACTIVE(and(active(afterNth(mark(x0), mark(x1))), y1))
MARK(and(x0, x1)) → ACTIVE(and(x0, x1))
MARK(tail(snd(x0))) → ACTIVE(tail(active(snd(mark(x0)))))
MARK(U11(head(x0), y1, y2, y3)) → ACTIVE(U11(active(head(mark(x0))), y1, y2, y3))
MARK(U11(afterNth(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(afterNth(mark(x0), mark(x1))), y1, y2, y3))
ACTIVE(and(tt, X)) → MARK(X)
MARK(and(U11(x0, x1, x2, x3), y1)) → ACTIVE(and(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(splitAt(y0, tt)) → ACTIVE(splitAt(mark(y0), active(tt)))
MARK(splitAt(tt, y1)) → ACTIVE(splitAt(active(tt), mark(y1)))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(U12(fst(x0), y1)) → ACTIVE(U12(active(fst(mark(x0))), y1))
MARK(s(X)) → MARK(X)
MARK(and(sel(x0, x1), y1)) → ACTIVE(and(active(sel(mark(x0), mark(x1))), y1))
MARK(head(tt)) → ACTIVE(head(active(tt)))
MARK(U11(cons(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1, y2, y3))
MARK(fst(sel(x0, x1))) → ACTIVE(fst(active(sel(mark(x0), mark(x1)))))
MARK(and(0, y1)) → ACTIVE(and(active(0), y1))
MARK(and(y0, mark(x1))) → ACTIVE(and(mark(y0), x1))
ACTIVE(U11(tt, N, X, XS)) → MARK(U12(splitAt(N, XS), X))
MARK(and(X1, X2)) → MARK(X1)
MARK(splitAt(y0, head(x0))) → ACTIVE(splitAt(mark(y0), active(head(mark(x0)))))
MARK(splitAt(y0, tail(x0))) → ACTIVE(splitAt(mark(y0), active(tail(mark(x0)))))
MARK(splitAt(U11(x0, x1, x2, x3), y1)) → ACTIVE(splitAt(active(U11(mark(x0), x1, x2, x3)), mark(y1)))
MARK(splitAt(y0, nil)) → ACTIVE(splitAt(mark(y0), active(nil)))
MARK(splitAt(nil, y1)) → ACTIVE(splitAt(active(nil), mark(y1)))
MARK(splitAt(y0, x1)) → ACTIVE(splitAt(mark(y0), x1))
MARK(U12(natsFrom(x0), y1)) → ACTIVE(U12(active(natsFrom(mark(x0))), y1))
MARK(U11(take(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1, y2, y3))
MARK(splitAt(y0, cons(x0, x1))) → ACTIVE(splitAt(mark(y0), active(cons(mark(x0), x1))))
MARK(splitAt(snd(x0), y1)) → ACTIVE(splitAt(active(snd(mark(x0))), mark(y1)))
MARK(fst(pair(x0, x1))) → ACTIVE(fst(active(pair(mark(x0), mark(x1)))))
MARK(and(head(x0), y1)) → ACTIVE(and(active(head(mark(x0))), y1))
MARK(U12(tail(x0), y1)) → ACTIVE(U12(active(tail(mark(x0))), y1))
MARK(U11(X1, X2, X3, X4)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
ACTIVE(tail(cons(N, XS))) → MARK(XS)
MARK(U11(y0, x1, x2, mark(x3))) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U11(0, y1, y2, y3)) → ACTIVE(U11(active(0), y1, y2, y3))
MARK(tail(pair(x0, x1))) → ACTIVE(tail(active(pair(mark(x0), mark(x1)))))
MARK(head(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U11(U11(x0, x1, x2, x3), y1, y2, y3)) → ACTIVE(U11(active(U11(mark(x0), x1, x2, x3)), y1, y2, y3))
MARK(splitAt(and(x0, x1), y1)) → ACTIVE(splitAt(active(and(mark(x0), x1)), mark(y1)))
ACTIVE(afterNth(N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(fst(fst(x0))) → ACTIVE(fst(active(fst(mark(x0)))))
ACTIVE(snd(pair(X, Y))) → MARK(Y)
MARK(splitAt(pair(x0, x1), y1)) → ACTIVE(splitAt(active(pair(mark(x0), mark(x1))), mark(y1)))
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(splitAt(take(x0, x1), y1)) → ACTIVE(splitAt(active(take(mark(x0), mark(x1))), mark(y1)))
MARK(and(y0, active(x1))) → ACTIVE(and(mark(y0), x1))
MARK(splitAt(U12(x0, x1), y1)) → ACTIVE(splitAt(active(U12(mark(x0), x1)), mark(y1)))
MARK(fst(U12(x0, x1))) → ACTIVE(fst(active(U12(mark(x0), x1))))
MARK(tail(afterNth(x0, x1))) → ACTIVE(tail(active(afterNth(mark(x0), mark(x1)))))
MARK(pair(X1, X2)) → MARK(X1)
MARK(U12(snd(x0), y1)) → ACTIVE(U12(active(snd(mark(x0))), y1))
MARK(fst(nil)) → ACTIVE(fst(active(nil)))
MARK(and(tail(x0), y1)) → ACTIVE(and(active(tail(mark(x0))), y1))
MARK(fst(take(x0, x1))) → ACTIVE(fst(active(take(mark(x0), mark(x1)))))
MARK(fst(x0)) → ACTIVE(fst(x0))
MARK(and(take(x0, x1), y1)) → ACTIVE(and(active(take(mark(x0), mark(x1))), y1))
MARK(U11(fst(x0), y1, y2, y3)) → ACTIVE(U11(active(fst(mark(x0))), y1, y2, y3))
MARK(pair(X1, X2)) → MARK(X2)
MARK(fst(U11(x0, x1, x2, x3))) → ACTIVE(fst(active(U11(mark(x0), x1, x2, x3))))
MARK(splitAt(y0, s(x0))) → ACTIVE(splitAt(mark(y0), active(s(mark(x0)))))
MARK(and(splitAt(x0, x1), y1)) → ACTIVE(and(active(splitAt(mark(x0), mark(x1))), y1))
MARK(snd(X)) → MARK(X)
MARK(head(sel(x0, x1))) → ACTIVE(head(active(sel(mark(x0), mark(x1)))))
MARK(splitAt(cons(x0, x1), y1)) → ACTIVE(splitAt(active(cons(mark(x0), x1)), mark(y1)))
MARK(splitAt(s(x0), y1)) → ACTIVE(splitAt(active(s(mark(x0))), mark(y1)))
MARK(U11(snd(x0), y1, y2, y3)) → ACTIVE(U11(active(snd(mark(x0))), y1, y2, y3))
MARK(U11(s(x0), y1, y2, y3)) → ACTIVE(U11(active(s(mark(x0))), y1, y2, y3))
MARK(U12(splitAt(x0, x1), y1)) → ACTIVE(U12(active(splitAt(mark(x0), mark(x1))), y1))
MARK(splitAt(afterNth(x0, x1), y1)) → ACTIVE(splitAt(active(afterNth(mark(x0), mark(x1))), mark(y1)))
MARK(U11(nil, y1, y2, y3)) → ACTIVE(U11(active(nil), y1, y2, y3))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))

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.
By narrowing [15] the rule MARK(snd(X)) → ACTIVE(snd(mark(X))) at position [0] we obtained the following new rules:

MARK(snd(x0)) → ACTIVE(snd(x0))
MARK(snd(U11(x0, x1, x2, x3))) → ACTIVE(snd(active(U11(mark(x0), x1, x2, x3))))
MARK(snd(snd(x0))) → ACTIVE(snd(active(snd(mark(x0)))))
MARK(snd(0)) → ACTIVE(snd(active(0)))
MARK(snd(splitAt(x0, x1))) → ACTIVE(snd(active(splitAt(mark(x0), mark(x1)))))
MARK(snd(tt)) → ACTIVE(snd(active(tt)))
MARK(snd(s(x0))) → ACTIVE(snd(active(s(mark(x0)))))
MARK(snd(U12(x0, x1))) → ACTIVE(snd(active(U12(mark(x0), x1))))
MARK(snd(nil)) → ACTIVE(snd(active(nil)))
MARK(snd(sel(x0, x1))) → ACTIVE(snd(active(sel(mark(x0), mark(x1)))))
MARK(snd(pair(x0, x1))) → ACTIVE(snd(active(pair(mark(x0), mark(x1)))))
MARK(snd(head(x0))) → ACTIVE(snd(active(head(mark(x0)))))
MARK(snd(natsFrom(x0))) → ACTIVE(snd(active(natsFrom(mark(x0)))))
MARK(snd(tail(x0))) → ACTIVE(snd(active(tail(mark(x0)))))
MARK(snd(and(x0, x1))) → ACTIVE(snd(active(and(mark(x0), x1))))
MARK(snd(cons(x0, x1))) → ACTIVE(snd(active(cons(mark(x0), x1))))
MARK(snd(fst(x0))) → ACTIVE(snd(active(fst(mark(x0)))))
MARK(snd(take(x0, x1))) → ACTIVE(snd(active(take(mark(x0), mark(x1)))))
MARK(snd(afterNth(x0, x1))) → ACTIVE(snd(active(afterNth(mark(x0), mark(x1)))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

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

MARK(U11(natsFrom(x0), y1, y2, y3)) → ACTIVE(U11(active(natsFrom(mark(x0))), y1, y2, y3))
MARK(snd(snd(x0))) → ACTIVE(snd(active(snd(mark(x0)))))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), y1))
MARK(head(snd(x0))) → ACTIVE(head(active(snd(mark(x0)))))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U11(tt, N, X, XS))
MARK(and(pair(x0, x1), y1)) → ACTIVE(and(active(pair(mark(x0), mark(x1))), y1))
MARK(U11(y0, mark(x1), x2, x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
ACTIVE(take(N, XS)) → MARK(fst(splitAt(N, XS)))
ACTIVE(sel(N, XS)) → MARK(head(afterNth(N, XS)))
MARK(snd(head(x0))) → ACTIVE(snd(active(head(mark(x0)))))
MARK(and(nil, y1)) → ACTIVE(and(active(nil), y1))
MARK(fst(head(x0))) → ACTIVE(fst(active(head(mark(x0)))))
MARK(splitAt(y0, take(x0, x1))) → ACTIVE(splitAt(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(U11(splitAt(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(splitAt(mark(x0), mark(x1))), y1, y2, y3))
MARK(U12(pair(x0, x1), y1)) → ACTIVE(U12(active(pair(mark(x0), mark(x1))), y1))
MARK(splitAt(y0, U11(x0, x1, x2, x3))) → ACTIVE(splitAt(mark(y0), active(U11(mark(x0), x1, x2, x3))))
MARK(U11(y0, x1, active(x2), x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(head(x0)) → ACTIVE(head(x0))
MARK(splitAt(y0, snd(x0))) → ACTIVE(splitAt(mark(y0), active(snd(mark(x0)))))
MARK(fst(cons(x0, x1))) → ACTIVE(fst(active(cons(mark(x0), x1))))
MARK(splitAt(y0, afterNth(x0, x1))) → ACTIVE(splitAt(mark(y0), active(afterNth(mark(x0), mark(x1)))))
MARK(head(U11(x0, x1, x2, x3))) → ACTIVE(head(active(U11(mark(x0), x1, x2, x3))))
MARK(tail(nil)) → ACTIVE(tail(active(nil)))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(head(head(x0))) → ACTIVE(head(active(head(mark(x0)))))
MARK(splitAt(y0, and(x0, x1))) → ACTIVE(splitAt(mark(y0), active(and(mark(x0), x1))))
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
MARK(U12(U11(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U11(mark(x0), x1, x2, x3)), y1))
MARK(tail(head(x0))) → ACTIVE(tail(active(head(mark(x0)))))
MARK(U12(sel(x0, x1), y1)) → ACTIVE(U12(active(sel(mark(x0), mark(x1))), y1))
MARK(take(X1, X2)) → MARK(X2)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
MARK(and(snd(x0), y1)) → ACTIVE(and(active(snd(mark(x0))), y1))
MARK(splitAt(y0, pair(x0, x1))) → ACTIVE(splitAt(mark(y0), active(pair(mark(x0), mark(x1)))))
ACTIVE(U12(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(fst(0)) → ACTIVE(fst(active(0)))
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(tail(cons(x0, x1))) → ACTIVE(tail(active(cons(mark(x0), x1))))
MARK(tail(x0)) → ACTIVE(tail(x0))
MARK(U11(pair(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(pair(mark(x0), mark(x1))), y1, y2, y3))
MARK(snd(x0)) → ACTIVE(snd(x0))
MARK(fst(snd(x0))) → ACTIVE(fst(active(snd(mark(x0)))))
MARK(head(U12(x0, x1))) → ACTIVE(head(active(U12(mark(x0), x1))))
MARK(tail(sel(x0, x1))) → ACTIVE(tail(active(sel(mark(x0), mark(x1)))))
MARK(tail(U12(x0, x1))) → ACTIVE(tail(active(U12(mark(x0), x1))))
MARK(fst(s(x0))) → ACTIVE(fst(active(s(mark(x0)))))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(U11(tt, y1, y2, y3)) → ACTIVE(U11(active(tt), y1, y2, y3))
MARK(tail(U11(x0, x1, x2, x3))) → ACTIVE(tail(active(U11(mark(x0), x1, x2, x3))))
MARK(head(natsFrom(x0))) → ACTIVE(head(active(natsFrom(mark(x0)))))
MARK(splitAt(0, y1)) → ACTIVE(splitAt(active(0), mark(y1)))
MARK(splitAt(y0, 0)) → ACTIVE(splitAt(mark(y0), active(0)))
MARK(tail(tt)) → ACTIVE(tail(active(tt)))
MARK(head(fst(x0))) → ACTIVE(head(active(fst(mark(x0)))))
MARK(U11(x0, x1, x2, x3)) → ACTIVE(U11(x0, x1, x2, x3))
MARK(snd(tt)) → ACTIVE(snd(active(tt)))
MARK(U12(afterNth(x0, x1), y1)) → ACTIVE(U12(active(afterNth(mark(x0), mark(x1))), y1))
MARK(and(U12(x0, x1), y1)) → ACTIVE(and(active(U12(mark(x0), x1)), y1))
MARK(splitAt(y0, natsFrom(x0))) → ACTIVE(splitAt(mark(y0), active(natsFrom(mark(x0)))))
MARK(take(X1, X2)) → MARK(X1)
MARK(U12(and(x0, x1), y1)) → ACTIVE(U12(active(and(mark(x0), x1)), y1))
MARK(fst(natsFrom(x0))) → ACTIVE(fst(active(natsFrom(mark(x0)))))
MARK(fst(splitAt(x0, x1))) → ACTIVE(fst(active(splitAt(mark(x0), mark(x1)))))
MARK(sel(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(head(splitAt(x0, x1))) → ACTIVE(head(active(splitAt(mark(x0), mark(x1)))))
MARK(snd(sel(x0, x1))) → ACTIVE(snd(active(sel(mark(x0), mark(x1)))))
MARK(fst(afterNth(x0, x1))) → ACTIVE(fst(active(afterNth(mark(x0), mark(x1)))))
MARK(U11(y0, active(x1), x2, x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
ACTIVE(fst(pair(X, Y))) → MARK(X)
MARK(head(pair(x0, x1))) → ACTIVE(head(active(pair(mark(x0), mark(x1)))))
MARK(splitAt(y0, fst(x0))) → ACTIVE(splitAt(mark(y0), active(fst(mark(x0)))))
MARK(tail(s(x0))) → ACTIVE(tail(active(s(mark(x0)))))
MARK(tail(0)) → ACTIVE(tail(active(0)))
MARK(afterNth(X1, X2)) → MARK(X1)
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(tail(fst(x0))) → ACTIVE(tail(active(fst(mark(x0)))))
MARK(fst(tail(x0))) → ACTIVE(fst(active(tail(mark(x0)))))
MARK(splitAt(x0, y1)) → ACTIVE(splitAt(x0, mark(y1)))
MARK(U11(sel(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(sel(mark(x0), mark(x1))), y1, y2, y3))
MARK(snd(0)) → ACTIVE(snd(active(0)))
MARK(and(natsFrom(x0), y1)) → ACTIVE(and(active(natsFrom(mark(x0))), y1))
MARK(splitAt(fst(x0), y1)) → ACTIVE(splitAt(active(fst(mark(x0))), mark(y1)))
MARK(U12(head(x0), y1)) → ACTIVE(U12(active(head(mark(x0))), y1))
MARK(natsFrom(X)) → MARK(X)
MARK(splitAt(y0, sel(x0, x1))) → ACTIVE(splitAt(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(tail(natsFrom(x0))) → ACTIVE(tail(active(natsFrom(mark(x0)))))
MARK(and(fst(x0), y1)) → ACTIVE(and(active(fst(mark(x0))), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(splitAt(tail(x0), y1)) → ACTIVE(splitAt(active(tail(mark(x0))), mark(y1)))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(U11(U12(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1, y2, y3))
MARK(tail(tail(x0))) → ACTIVE(tail(active(tail(mark(x0)))))
MARK(U11(y0, x1, mark(x2), x3)) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U11(y0, x1, x2, active(x3))) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(head(s(x0))) → ACTIVE(head(active(s(mark(x0)))))
MARK(splitAt(splitAt(x0, x1), y1)) → ACTIVE(splitAt(active(splitAt(mark(x0), mark(x1))), mark(y1)))
MARK(U11(tail(x0), y1, y2, y3)) → ACTIVE(U11(active(tail(mark(x0))), y1, y2, y3))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), y1))
MARK(splitAt(head(x0), y1)) → ACTIVE(splitAt(active(head(mark(x0))), mark(y1)))
MARK(tail(splitAt(x0, x1))) → ACTIVE(tail(active(splitAt(mark(x0), mark(x1)))))
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(snd(tail(x0))) → ACTIVE(snd(active(tail(mark(x0)))))
MARK(snd(fst(x0))) → ACTIVE(snd(active(fst(mark(x0)))))
MARK(head(take(x0, x1))) → ACTIVE(head(active(take(mark(x0), mark(x1)))))
MARK(splitAt(y0, U12(x0, x1))) → ACTIVE(splitAt(mark(y0), active(U12(mark(x0), x1))))
MARK(head(nil)) → ACTIVE(head(active(nil)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), x1)), y1))
MARK(U11(and(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(and(mark(x0), x1)), y1, y2, y3))
MARK(head(0)) → ACTIVE(head(active(0)))
MARK(head(tail(x0))) → ACTIVE(head(active(tail(mark(x0)))))
MARK(splitAt(natsFrom(x0), y1)) → ACTIVE(splitAt(active(natsFrom(mark(x0))), mark(y1)))
MARK(head(afterNth(x0, x1))) → ACTIVE(head(active(afterNth(mark(x0), mark(x1)))))
MARK(head(cons(x0, x1))) → ACTIVE(head(active(cons(mark(x0), x1))))
MARK(tail(and(x0, x1))) → ACTIVE(tail(active(and(mark(x0), x1))))
MARK(fst(X)) → MARK(X)
MARK(snd(afterNth(x0, x1))) → ACTIVE(snd(active(afterNth(mark(x0), mark(x1)))))
MARK(fst(tt)) → ACTIVE(fst(active(tt)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(fst(and(x0, x1))) → ACTIVE(fst(active(and(mark(x0), x1))))
MARK(head(and(x0, x1))) → ACTIVE(head(active(and(mark(x0), x1))))
MARK(splitAt(y0, splitAt(x0, x1))) → ACTIVE(splitAt(mark(y0), active(splitAt(mark(x0), mark(x1)))))
MARK(tail(take(x0, x1))) → ACTIVE(tail(active(take(mark(x0), mark(x1)))))
MARK(splitAt(sel(x0, x1), y1)) → ACTIVE(splitAt(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(and(afterNth(x0, x1), y1)) → ACTIVE(and(active(afterNth(mark(x0), mark(x1))), y1))
MARK(and(x0, x1)) → ACTIVE(and(x0, x1))
MARK(tail(snd(x0))) → ACTIVE(tail(active(snd(mark(x0)))))
MARK(U11(head(x0), y1, y2, y3)) → ACTIVE(U11(active(head(mark(x0))), y1, y2, y3))
MARK(U11(afterNth(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(afterNth(mark(x0), mark(x1))), y1, y2, y3))
ACTIVE(and(tt, X)) → MARK(X)
MARK(and(U11(x0, x1, x2, x3), y1)) → ACTIVE(and(active(U11(mark(x0), x1, x2, x3)), y1))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(splitAt(tt, y1)) → ACTIVE(splitAt(active(tt), mark(y1)))
MARK(splitAt(y0, tt)) → ACTIVE(splitAt(mark(y0), active(tt)))
MARK(U12(fst(x0), y1)) → ACTIVE(U12(active(fst(mark(x0))), y1))
MARK(s(X)) → MARK(X)
MARK(snd(splitAt(x0, x1))) → ACTIVE(snd(active(splitAt(mark(x0), mark(x1)))))
MARK(head(tt)) → ACTIVE(head(active(tt)))
MARK(and(sel(x0, x1), y1)) → ACTIVE(and(active(sel(mark(x0), mark(x1))), y1))
MARK(snd(s(x0))) → ACTIVE(snd(active(s(mark(x0)))))
MARK(U11(cons(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1, y2, y3))
MARK(snd(pair(x0, x1))) → ACTIVE(snd(active(pair(mark(x0), mark(x1)))))
MARK(fst(sel(x0, x1))) → ACTIVE(fst(active(sel(mark(x0), mark(x1)))))
MARK(and(0, y1)) → ACTIVE(and(active(0), y1))
MARK(and(y0, mark(x1))) → ACTIVE(and(mark(y0), x1))
ACTIVE(U11(tt, N, X, XS)) → MARK(U12(splitAt(N, XS), X))
MARK(snd(natsFrom(x0))) → ACTIVE(snd(active(natsFrom(mark(x0)))))
MARK(and(X1, X2)) → MARK(X1)
MARK(splitAt(y0, tail(x0))) → ACTIVE(splitAt(mark(y0), active(tail(mark(x0)))))
MARK(splitAt(y0, head(x0))) → ACTIVE(splitAt(mark(y0), active(head(mark(x0)))))
MARK(splitAt(U11(x0, x1, x2, x3), y1)) → ACTIVE(splitAt(active(U11(mark(x0), x1, x2, x3)), mark(y1)))
MARK(snd(cons(x0, x1))) → ACTIVE(snd(active(cons(mark(x0), x1))))
MARK(splitAt(nil, y1)) → ACTIVE(splitAt(active(nil), mark(y1)))
MARK(splitAt(y0, nil)) → ACTIVE(splitAt(mark(y0), active(nil)))
MARK(snd(take(x0, x1))) → ACTIVE(snd(active(take(mark(x0), mark(x1)))))
MARK(splitAt(y0, x1)) → ACTIVE(splitAt(mark(y0), x1))
MARK(U12(natsFrom(x0), y1)) → ACTIVE(U12(active(natsFrom(mark(x0))), y1))
MARK(snd(nil)) → ACTIVE(snd(active(nil)))
MARK(U11(take(x0, x1), y1, y2, y3)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1, y2, y3))
MARK(splitAt(y0, cons(x0, x1))) → ACTIVE(splitAt(mark(y0), active(cons(mark(x0), x1))))
MARK(splitAt(snd(x0), y1)) → ACTIVE(splitAt(active(snd(mark(x0))), mark(y1)))
MARK(fst(pair(x0, x1))) → ACTIVE(fst(active(pair(mark(x0), mark(x1)))))
MARK(and(head(x0), y1)) → ACTIVE(and(active(head(mark(x0))), y1))
MARK(U12(tail(x0), y1)) → ACTIVE(U12(active(tail(mark(x0))), y1))
MARK(U11(X1, X2, X3, X4)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
ACTIVE(tail(cons(N, XS))) → MARK(XS)
MARK(U11(y0, x1, x2, mark(x3))) → ACTIVE(U11(mark(y0), x1, x2, x3))
MARK(U11(0, y1, y2, y3)) → ACTIVE(U11(active(0), y1, y2, y3))
MARK(tail(pair(x0, x1))) → ACTIVE(tail(active(pair(mark(x0), mark(x1)))))
MARK(tail(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U11(U11(x0, x1, x2, x3), y1, y2, y3)) → ACTIVE(U11(active(U11(mark(x0), x1, x2, x3)), y1, y2, y3))
MARK(splitAt(and(x0, x1), y1)) → ACTIVE(splitAt(active(and(mark(x0), x1)), mark(y1)))
ACTIVE(afterNth(N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(fst(fst(x0))) → ACTIVE(fst(active(fst(mark(x0)))))
ACTIVE(snd(pair(X, Y))) → MARK(Y)
MARK(snd(and(x0, x1))) → ACTIVE(snd(active(and(mark(x0), x1))))
MARK(splitAt(pair(x0, x1), y1)) → ACTIVE(splitAt(active(pair(mark(x0), mark(x1))), mark(y1)))
MARK(splitAt(take(x0, x1), y1)) → ACTIVE(splitAt(active(take(mark(x0), mark(x1))), mark(y1)))
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(and(y0, active(x1))) → ACTIVE(and(mark(y0), x1))
MARK(splitAt(U12(x0, x1), y1)) → ACTIVE(splitAt(active(U12(mark(x0), x1)), mark(y1)))
MARK(fst(U12(x0, x1))) → ACTIVE(fst(active(U12(mark(x0), x1))))
MARK(pair(X1, X2)) → MARK(X1)
MARK(tail(afterNth(x0, x1))) → ACTIVE(tail(active(afterNth(mark(x0), mark(x1)))))
MARK(snd(U12(x0, x1))) → ACTIVE(snd(active(U12(mark(x0), x1))))
MARK(U12(snd(x0), y1)) → ACTIVE(U12(active(snd(mark(x0))), y1))
MARK(fst(nil)) → ACTIVE(fst(active(nil)))
MARK(and(tail(x0), y1)) → ACTIVE(and(active(tail(mark(x0))), y1))
MARK(fst(take(x0, x1))) → ACTIVE(fst(active(take(mark(x0), mark(x1)))))
MARK(fst(x0)) → ACTIVE(fst(x0))
MARK(and(take(x0, x1), y1)) → ACTIVE(and(active(take(mark(x0), mark(x1))), y1))
MARK(U11(fst(x0), y1, y2, y3)) → ACTIVE(U11(active(fst(mark(x0))), y1, y2, y3))
MARK(pair(X1, X2)) → MARK(X2)
MARK(fst(U11(x0, x1, x2, x3))) → ACTIVE(fst(active(U11(mark(x0), x1, x2, x3))))
MARK(splitAt(y0, s(x0))) → ACTIVE(splitAt(mark(y0), active(s(mark(x0)))))
MARK(and(splitAt(x0, x1), y1)) → ACTIVE(and(active(splitAt(mark(x0), mark(x1))), y1))
MARK(snd(X)) → MARK(X)
MARK(snd(U11(x0, x1, x2, x3))) → ACTIVE(snd(active(U11(mark(x0), x1, x2, x3))))
MARK(head(sel(x0, x1))) → ACTIVE(head(active(sel(mark(x0), mark(x1)))))
MARK(splitAt(cons(x0, x1), y1)) → ACTIVE(splitAt(active(cons(mark(x0), x1)), mark(y1)))
MARK(splitAt(s(x0), y1)) → ACTIVE(splitAt(active(s(mark(x0))), mark(y1)))
MARK(U11(snd(x0), y1, y2, y3)) → ACTIVE(U11(active(snd(mark(x0))), y1, y2, y3))
MARK(U12(splitAt(x0, x1), y1)) → ACTIVE(U12(active(splitAt(mark(x0), mark(x1))), y1))
MARK(U11(s(x0), y1, y2, y3)) → ACTIVE(U11(active(s(mark(x0))), y1, y2, y3))
MARK(splitAt(afterNth(x0, x1), y1)) → ACTIVE(splitAt(active(afterNth(mark(x0), mark(x1))), mark(y1)))
MARK(U11(nil, y1, y2, y3)) → ACTIVE(U11(active(nil), y1, y2, y3))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))

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.