Termination w.r.t. Q proof of /tmp/tmpcDoEDJ/ExSec4_2_DLMMU04

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

The TRS R consists of the following rules:

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

The TRS R consists of the following rules:

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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 14 SCCs with 29 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ 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(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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.

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 remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(TAKE(x₁, x₂)) = x_1 + (2)x_2
POL(active(x₁)) = 1 + (4)x_1
POL(mark(x₁)) = 1 + (2)x_1

The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ 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(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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.

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 remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x₁)) = 1 + (4)x_1
POL(mark(x₁)) = 1 + (2)x_1
POL(AFTERNTH(x₁, x₂)) = x_1 + (2)x_2

The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ 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(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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.

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 remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x₁)) = 1 + (2)x_1
POL(mark(x₁)) = 1 + (4)x_1
POL(SEL(x₁, x₂)) = x_1 + (2)x_2

The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ 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(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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.

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

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x₁)) = 4 + x_1
POL(mark(x₁)) = 4 + (4)x_1
POL(TAIL(x₁)) = (4)x_1

The value of delta used in the strict ordering is 16.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ 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(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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.

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

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x₁)) = 4 + (4)x_1
POL(mark(x₁)) = 4 + x_1
POL(HEAD(x₁)) = (4)x_1

The value of delta used in the strict ordering is 16.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

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

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(U(x₁, x₂, x₃, x₄)) = (2)x_1 + x_2 + (3)x_3 + (3)x_4
POL(active(x₁)) = 3 + (4)x_1
POL(mark(x₁)) = 1 + (2)x_1

The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ 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(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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.

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 remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x₁)) = 4 + (2)x_1
POL(SPLITAT(x₁, x₂)) = (4)x_1 + (4)x_2
POL(mark(x₁)) = 4 + (2)x_1

The value of delta used in the strict ordering is 16.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ 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(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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.

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

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x₁)) = 4 + (4)x_1
POL(mark(x₁)) = 4 + x_1
POL(SND(x₁)) = (4)x_1

The value of delta used in the strict ordering is 16.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ 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(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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.

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 remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x₁)) = 4 + (4)x_1
POL(PAIR(x₁, x₂)) = (3)x_1 + (4)x_2
POL(mark(x₁)) = 4 + (4)x_1

The value of delta used in the strict ordering is 12.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ 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(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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.

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

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x₁)) = 4 + x_1
POL(FST(x₁)) = (4)x_1
POL(mark(x₁)) = 4 + (4)x_1

The value of delta used in the strict ordering is 16.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ 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(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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.

S(mark(X)) → S(X)
S(active(X)) → S(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x₁)) = 4 + x_1
POL(mark(x₁)) = 4 + (4)x_1
POL(S(x₁)) = (4)x_1

The value of delta used in the strict ordering is 16.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ 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(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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.

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 remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x₁)) = 1 + (4)x_1
POL(CONS(x₁, x₂)) = x_1 + (4)x_2
POL(mark(x₁)) = 3 + (4)x_1

The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

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

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ 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(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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.

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

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x₁)) = 4 + x_1
POL(mark(x₁)) = 4 + (4)x_1
POL(NATSFROM(x₁)) = (4)x_1

The value of delta used in the strict ordering is 16.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

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

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(fst(X)) → active(fst(mark(X)))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(u(X1, X2, X3, X4)) → active(u(mark(X1), X2, X3, X4))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
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)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
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)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
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)
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ 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(pair(X1, X2)) → ACTIVE(pair(mark(X1), mark(X2)))
MARK(s(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(u(X1, X2, X3, X4)) → ACTIVE(u(mark(X1), X2, X3, X4))
MARK(cons(X1, X2)) → MARK(X1)
MARK(natsFrom(X)) → MARK(X)
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(s(X)) → ACTIVE(s(mark(X)))
MARK(pair(X1, X2)) → MARK(X2)
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(u(splitAt(N, XS), N, X, XS))
MARK(take(X1, X2)) → MARK(X1)
ACTIVE(snd(pair(XS, YS))) → MARK(YS)
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)
ACTIVE(u(pair(YS, ZS), N, X, XS)) → MARK(pair(cons(X, YS), ZS))
MARK(snd(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(u(X1, X2, X3, X4)) → MARK(X1)
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(XS, YS))) → MARK(XS)
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(afterNth(X1, X2)) → MARK(X1)
ACTIVE(tail(cons(N, XS))) → MARK(XS)

The TRS R consists of the following rules:

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

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(u(X1, X2, X3, X4)) → ACTIVE(u(mark(X1), X2, X3, X4))
MARK(cons(X1, X2)) → MARK(X1)
MARK(natsFrom(X)) → MARK(X)
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(pair(X1, X2)) → MARK(X2)
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(u(splitAt(N, XS), N, X, XS))
MARK(take(X1, X2)) → MARK(X1)
ACTIVE(snd(pair(XS, YS))) → MARK(YS)
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)
ACTIVE(u(pair(YS, ZS), N, X, XS)) → MARK(pair(cons(X, YS), ZS))
MARK(snd(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(u(X1, X2, X3, X4)) → MARK(X1)
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(XS, YS))) → MARK(XS)
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(afterNth(X1, X2)) → MARK(X1)
ACTIVE(tail(cons(N, XS))) → MARK(XS)

Used ordering: Polynomial interpretation [25,35]:

POL(sel(x₁, x₂)) = 1
POL(afterNth(x₁, x₂)) = 1
POL(tail(x₁)) = 1
POL(snd(x₁)) = 1
POL(head(x₁)) = 1
POL(mark(x₁)) = 0
POL(take(x₁, x₂)) = 1
POL(splitAt(x₁, x₂)) = 1
POL(0) = 0
POL(ACTIVE(x₁)) = (4)x_1
POL(active(x₁)) = 2
POL(cons(x₁, x₂)) = 1
POL(MARK(x₁)) = 4
POL(u(x₁, x₂, x₃, x₄)) = 1
POL(pair(x₁, x₂)) = 0
POL(fst(x₁)) = 1
POL(s(x₁)) = 0
POL(natsFrom(x₁)) = 1
POL(nil) = 2

The value of delta used in the strict ordering is 4.
The following usable rules [17] were oriented:

natsFrom(active(X)) → natsFrom(X)
natsFrom(mark(X)) → natsFrom(X)
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)
snd(active(X)) → snd(X)
snd(mark(X)) → snd(X)
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)
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)
s(active(X)) → s(X)
s(mark(X)) → s(X)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(active(X)) → head(X)
head(mark(X)) → head(X)
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)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(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)
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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

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(u(X1, X2, X3, X4)) → ACTIVE(u(mark(X1), X2, X3, X4))
MARK(head(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(natsFrom(X)) → MARK(X)
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(pair(X1, X2)) → MARK(X2)
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(u(splitAt(N, XS), N, X, XS))
MARK(take(X1, X2)) → MARK(X1)
ACTIVE(snd(pair(XS, YS))) → MARK(YS)
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)
ACTIVE(u(pair(YS, ZS), N, X, XS)) → MARK(pair(cons(X, YS), ZS))
MARK(snd(X)) → MARK(X)
MARK(sel(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(u(X1, X2, X3, X4)) → MARK(X1)
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(XS, YS))) → MARK(XS)
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(sel(X1, X2)) → MARK(X2)
ACTIVE(tail(cons(N, XS))) → MARK(XS)
MARK(afterNth(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

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

The remaining pairs can at least be oriented weakly.

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(u(X1, X2, X3, X4)) → ACTIVE(u(mark(X1), X2, X3, X4))
MARK(head(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(natsFrom(X)) → MARK(X)
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(pair(X1, X2)) → MARK(X2)
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(u(splitAt(N, XS), N, X, XS))
MARK(take(X1, X2)) → MARK(X1)
ACTIVE(snd(pair(XS, YS))) → MARK(YS)
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)
ACTIVE(u(pair(YS, ZS), N, X, XS)) → MARK(pair(cons(X, YS), ZS))
MARK(snd(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(u(X1, X2, X3, X4)) → MARK(X1)
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(XS, YS))) → MARK(XS)
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(sel(X1, X2)) → MARK(X2)
ACTIVE(tail(cons(N, XS))) → MARK(XS)
MARK(afterNth(X1, X2)) → MARK(X1)

Used ordering: Polynomial interpretation [25,35]:

POL(sel(x₁, x₂)) = 1
POL(tail(x₁)) = 1
POL(afterNth(x₁, x₂)) = 1
POL(snd(x₁)) = 1
POL(head(x₁)) = 1
POL(mark(x₁)) = 0
POL(take(x₁, x₂)) = 1
POL(splitAt(x₁, x₂)) = 1
POL(0) = 2
POL(ACTIVE(x₁)) = (2)x_1
POL(active(x₁)) = 3 + (4)x_1
POL(cons(x₁, x₂)) = 0
POL(MARK(x₁)) = 2
POL(u(x₁, x₂, x₃, x₄)) = 1
POL(pair(x₁, x₂)) = (4)x_2
POL(fst(x₁)) = 1
POL(s(x₁)) = 3
POL(nil) = 1
POL(natsFrom(x₁)) = 1

The value of delta used in the strict ordering is 2.
The following usable rules [17] were oriented:

natsFrom(active(X)) → natsFrom(X)
natsFrom(mark(X)) → natsFrom(X)
snd(active(X)) → snd(X)
snd(mark(X)) → snd(X)
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)
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)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
head(active(X)) → head(X)
head(mark(X)) → head(X)
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)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
u(X1, X2, mark(X3), X4) → u(X1, X2, X3, X4)
u(X1, active(X2), X3, X4) → u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, active(X4)) → u(X1, X2, X3, X4)
u(X1, mark(X2), X3, X4) → u(X1, X2, X3, X4)
u(mark(X1), X2, X3, X4) → u(X1, X2, X3, X4)
u(X1, X2, X3, mark(X4)) → u(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)
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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

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(tail(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(u(X1, X2, X3, X4)) → ACTIVE(u(mark(X1), X2, X3, X4))
MARK(cons(X1, X2)) → MARK(X1)
MARK(natsFrom(X)) → MARK(X)
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(pair(X1, X2)) → MARK(X2)
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(u(splitAt(N, XS), N, X, XS))
MARK(take(X1, X2)) → MARK(X1)
ACTIVE(snd(pair(XS, YS))) → MARK(YS)
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)
ACTIVE(u(pair(YS, ZS), N, X, XS)) → MARK(pair(cons(X, YS), ZS))
MARK(snd(X)) → MARK(X)
MARK(sel(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(u(X1, X2, X3, X4)) → MARK(X1)
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(XS, YS))) → MARK(XS)
ACTIVE(head(cons(N, XS))) → MARK(N)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(afterNth(X1, X2)) → MARK(X1)
ACTIVE(tail(cons(N, XS))) → MARK(XS)

The TRS R consists of the following rules:

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