Termination w.r.t. Q proof of /tmp/tmp6QZvHX/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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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:

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

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:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 16 SCCs with 47 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

TAKE(ok(X1), ok(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(ok(X1), ok(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)) = (4)x_1 + (4)x_2
POL(ok(x₁)) = 4 + (2)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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

AFTERNTH(ok(X1), ok(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(ok(X1), ok(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(ok(x₁)) = 4 + (2)x_1
POL(mark(x₁)) = 4 + (2)x_1
POL(AFTERNTH(x₁, x₂)) = (4)x_1 + (4)x_2

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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

SEL(mark(X1), X2) → SEL(X1, X2)
SEL(ok(X1), ok(X2)) → SEL(X1, X2)
SEL(X1, mark(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

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

POL(mark(x₁)) = 1 + (4)x_1
POL(ok(x₁)) = 3 + (4)x_1
POL(SEL(x₁, x₂)) = (4)x_1 + 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

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

TAIL(ok(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(ok(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(ok(x₁)) = 4 + (4)x_1
POL(mark(x₁)) = 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

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

HEAD(mark(X)) → HEAD(X)
HEAD(ok(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(ok(X)) → HEAD(X)

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

POL(mark(x₁)) = 4 + (4)x_1
POL(ok(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

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

U(mark(X1), X2, X3, X4) → U(X1, X2, X3, X4)
U(ok(X1), ok(X2), ok(X3), ok(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(mark(X1), X2, X3, X4) → U(X1, X2, X3, X4)
U(ok(X1), ok(X2), ok(X3), ok(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 + x_4
POL(mark(x₁)) = 2 + x_1
POL(ok(x₁)) = 1 + (2)x_1

The value of delta used in the strict ordering is 4.
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

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ 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(ok(X1), ok(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(ok(X1), ok(X2)) → SPLITAT(X1, X2)

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

POL(SPLITAT(x₁, x₂)) = (4)x_1 + x_2
POL(mark(x₁)) = 3 + (4)x_1
POL(ok(x₁)) = 1 + (4)x_1

The value of delta used in the strict ordering is 3.
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

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

SND(mark(X)) → SND(X)
SND(ok(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(ok(X)) → SND(X)

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

POL(mark(x₁)) = 4 + x_1
POL(ok(x₁)) = 4 + (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

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

PAIR(ok(X1), ok(X2)) → PAIR(X1, X2)
PAIR(mark(X1), X2) → PAIR(X1, X2)
PAIR(X1, mark(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

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

POL(PAIR(x₁, x₂)) = (4)x_1 + x_2
POL(ok(x₁)) = 1 + (4)x_1
POL(mark(x₁)) = 4 + (4)x_1

The value of delta used in the strict ordering is 4.
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

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

FST(ok(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(ok(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(FST(x₁)) = (4)x_1
POL(ok(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

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

S(ok(X)) → S(X)
S(mark(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(ok(X)) → S(X)
S(mark(X)) → S(X)

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

POL(ok(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

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(ok(X1), ok(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

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

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

The value of delta used in the strict ordering is 15.
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

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

NATSFROM(ok(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(ok(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(ok(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

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

PROPER(u(X1, X2, X3, X4)) → PROPER(X2)
PROPER(sel(X1, X2)) → PROPER(X2)
PROPER(pair(X1, X2)) → PROPER(X1)
PROPER(splitAt(X1, X2)) → PROPER(X2)
PROPER(pair(X1, X2)) → PROPER(X2)
PROPER(head(X)) → PROPER(X)
PROPER(natsFrom(X)) → PROPER(X)
PROPER(u(X1, X2, X3, X4)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(u(X1, X2, X3, X4)) → PROPER(X3)
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(tail(X)) → PROPER(X)
PROPER(splitAt(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(snd(X)) → PROPER(X)
PROPER(afterNth(X1, X2)) → PROPER(X1)
PROPER(afterNth(X1, X2)) → PROPER(X2)
PROPER(fst(X)) → PROPER(X)
PROPER(u(X1, X2, X3, X4)) → PROPER(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

PROPER(u(X1, X2, X3, X4)) → PROPER(X2)
PROPER(sel(X1, X2)) → PROPER(X2)
PROPER(pair(X1, X2)) → PROPER(X1)
PROPER(splitAt(X1, X2)) → PROPER(X2)
PROPER(pair(X1, X2)) → PROPER(X2)
PROPER(head(X)) → PROPER(X)
PROPER(natsFrom(X)) → PROPER(X)
PROPER(u(X1, X2, X3, X4)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(u(X1, X2, X3, X4)) → PROPER(X3)
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(tail(X)) → PROPER(X)
PROPER(splitAt(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(snd(X)) → PROPER(X)
PROPER(afterNth(X1, X2)) → PROPER(X1)
PROPER(afterNth(X1, X2)) → PROPER(X2)
PROPER(fst(X)) → PROPER(X)
PROPER(u(X1, X2, X3, X4)) → PROPER(X4)

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

POL(sel(x₁, x₂)) = 4 + (4)x_1 + (4)x_2
POL(PROPER(x₁)) = (4)x_1
POL(tail(x₁)) = 4 + (4)x_1
POL(afterNth(x₁, x₂)) = 4 + (4)x_1 + (4)x_2
POL(snd(x₁)) = 4 + (4)x_1
POL(head(x₁)) = 4 + (4)x_1
POL(take(x₁, x₂)) = 4 + (4)x_1 + (4)x_2
POL(splitAt(x₁, x₂)) = 4 + (4)x_1 + (4)x_2
POL(cons(x₁, x₂)) = 4 + (4)x_1 + (4)x_2
POL(u(x₁, x₂, x₃, x₄)) = 4 + (2)x_1 + (4)x_2 + (4)x_3 + (4)x_4
POL(fst(x₁)) = 4 + (4)x_1
POL(pair(x₁, x₂)) = 4 + (4)x_1 + (4)x_2
POL(s(x₁)) = 4 + x_1
POL(natsFrom(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

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

ACTIVE(pair(X1, X2)) → ACTIVE(X2)
ACTIVE(snd(X)) → ACTIVE(X)
ACTIVE(pair(X1, X2)) → ACTIVE(X1)
ACTIVE(natsFrom(X)) → ACTIVE(X)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(splitAt(X1, X2)) → ACTIVE(X2)
ACTIVE(afterNth(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X2)
ACTIVE(afterNth(X1, X2)) → ACTIVE(X2)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(splitAt(X1, X2)) → ACTIVE(X1)
ACTIVE(fst(X)) → ACTIVE(X)
ACTIVE(head(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(u(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(tail(X)) → ACTIVE(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

ACTIVE(pair(X1, X2)) → ACTIVE(X2)
ACTIVE(snd(X)) → ACTIVE(X)
ACTIVE(pair(X1, X2)) → ACTIVE(X1)
ACTIVE(natsFrom(X)) → ACTIVE(X)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(splitAt(X1, X2)) → ACTIVE(X2)
ACTIVE(afterNth(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X2)
ACTIVE(afterNth(X1, X2)) → ACTIVE(X2)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(splitAt(X1, X2)) → ACTIVE(X1)
ACTIVE(fst(X)) → ACTIVE(X)
ACTIVE(head(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(u(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(tail(X)) → ACTIVE(X)

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

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

The value of delta used in the strict ordering is 8.
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

          ↳ 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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