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

(0) Obligation:

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

active(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.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 16 SCCs with 47 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(X1, mark(X2)) → TAKE(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

[active₁, natsFrom₁, proper₁] > pair₂ > [cons₂, u₄] > mark₁
[active₁, natsFrom₁, proper₁] > splitAt₂ > [cons₂, u₄] > mark₁
[active₁, natsFrom₁, proper₁] > 0 > nil > mark₁
[active₁, natsFrom₁, proper₁] > sel₂ > afterNth₂ > mark₁
[active₁, natsFrom₁, proper₁] > take₂ > fst₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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))

(7) Obligation:

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

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

[proper, top] > [TAKE₁, mark, ok₁, 0, nil]

The following usable rules [FROCOS05] were oriented:

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))

(9) Obligation:

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

TAKE(mark(X1), 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.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(mark(X1), X2) → TAKE(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

TAKE₁ > [mark₁, fst₁, tail₁]
0 > [mark₁, fst₁, tail₁]
top > active₁ > natsFrom₁ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > pair₂ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > nil > [mark₁, fst₁, tail₁]
top > active₁ > u₂ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > sel₂ > [mark₁, fst₁, tail₁]
top > active₁ > afterNth₂ > snd₁ > [mark₁, fst₁, tail₁]
top > active₁ > afterNth₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]

The following usable rules [FROCOS05] were oriented:

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))

(11) Obligation:

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.

(12) PisEmptyProof (EQUIVALENT transformation)

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

(13) TRUE

(14) Obligation:

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

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Recursive Path Order [RPO].
Precedence:

[active₁, natsFrom₁, proper₁] > pair₂ > [cons₂, u₄] > mark₁
[active₁, natsFrom₁, proper₁] > splitAt₂ > [cons₂, u₄] > mark₁
[active₁, natsFrom₁, proper₁] > 0 > nil > mark₁
[active₁, natsFrom₁, proper₁] > sel₂ > afterNth₂ > mark₁
[active₁, natsFrom₁, proper₁] > take₂ > fst₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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))

(16) Obligation:

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

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AFTERNTH(ok(X1), ok(X2)) → AFTERNTH(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

[proper, top] > [AFTERNTH₁, mark, ok₁, 0, nil]

The following usable rules [FROCOS05] were oriented:

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))

(18) Obligation:

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

AFTERNTH(mark(X1), 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.

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Recursive Path Order [RPO].
Precedence:

AFTERNTH₁ > [mark₁, fst₁, tail₁]
0 > [mark₁, fst₁, tail₁]
top > active₁ > natsFrom₁ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > pair₂ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > nil > [mark₁, fst₁, tail₁]
top > active₁ > u₂ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > sel₂ > [mark₁, fst₁, tail₁]
top > active₁ > afterNth₂ > snd₁ > [mark₁, fst₁, tail₁]
top > active₁ > afterNth₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]

The following usable rules [FROCOS05] were oriented:

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))

(20) Obligation:

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.

(21) PisEmptyProof (EQUIVALENT transformation)

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

(22) TRUE

(23) Obligation:

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

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

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(X1, mark(X2)) → SEL(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

[active₁, natsFrom₁, proper₁] > pair₂ > [cons₂, u₄] > mark₁
[active₁, natsFrom₁, proper₁] > splitAt₂ > [cons₂, u₄] > mark₁
[active₁, natsFrom₁, proper₁] > 0 > nil > mark₁
[active₁, natsFrom₁, proper₁] > sel₂ > afterNth₂ > mark₁
[active₁, natsFrom₁, proper₁] > take₂ > fst₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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))

(25) Obligation:

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)

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.

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(ok(X1), ok(X2)) → SEL(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

[proper, top] > [SEL₁, mark, ok₁, 0, nil]

The following usable rules [FROCOS05] were oriented:

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))

(27) Obligation:

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

SEL(mark(X1), X2) → SEL(X1, X2)

The TRS R consists of the following rules:

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

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(mark(X1), X2) → SEL(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

SEL₁ > [mark₁, fst₁, tail₁]
0 > [mark₁, fst₁, tail₁]
top > active₁ > natsFrom₁ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > pair₂ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > nil > [mark₁, fst₁, tail₁]
top > active₁ > u₂ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > sel₂ > [mark₁, fst₁, tail₁]
top > active₁ > afterNth₂ > snd₁ > [mark₁, fst₁, tail₁]
top > active₁ > afterNth₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]

The following usable rules [FROCOS05] were oriented:

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))

(29) Obligation:

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.

(30) PisEmptyProof (EQUIVALENT transformation)

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

(31) TRUE

(32) Obligation:

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.

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(mark(X)) → TAIL(X)

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

Recursive Path Order [RPO].
Precedence:

TAIL₁ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > cons₂ > [splitAt₂, u₄] > pair₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > 0 > pair₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > nil > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > sel₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > afterNth₂ > snd₁ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > afterNth₂ > [splitAt₂, u₄] > pair₂ > [mark₁, natsFrom₁, fst₁]
top > [mark₁, natsFrom₁, fst₁]

The following usable rules [FROCOS05] were oriented:

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))

(34) Obligation:

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

TAIL(ok(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.

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(ok(X)) → TAIL(X)

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

Recursive Path Order [RPO].
Precedence:

[active₁, cons₂, nil, afterNth₂, take₁, proper₁] > [TAIL₁, ok₁, fst₁, splitAt₁, head₁] > top > mark
[active₁, cons₂, nil, afterNth₂, take₁, proper₁] > 0 > mark

The following usable rules [FROCOS05] were oriented:

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))

(36) Obligation:

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.

(37) PisEmptyProof (EQUIVALENT transformation)

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

(38) TRUE

(39) Obligation:

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

HEAD(ok(X)) → HEAD(X)
HEAD(mark(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.

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(mark(X)) → HEAD(X)

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

Recursive Path Order [RPO].
Precedence:

HEAD₁ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > cons₂ > [splitAt₂, u₄] > pair₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > 0 > pair₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > nil > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > sel₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > afterNth₂ > snd₁ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > afterNth₂ > [splitAt₂, u₄] > pair₂ > [mark₁, natsFrom₁, fst₁]
top > [mark₁, natsFrom₁, fst₁]

The following usable rules [FROCOS05] were oriented:

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))

(41) Obligation:

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

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.

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(ok(X)) → HEAD(X)

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

Recursive Path Order [RPO].
Precedence:

[active₁, cons₂, nil, afterNth₂, take₁, proper₁] > [HEAD₁, ok₁, fst₁, splitAt₁, head₁] > top > mark
[active₁, cons₂, nil, afterNth₂, take₁, proper₁] > 0 > mark

The following usable rules [FROCOS05] were oriented:

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))

(43) Obligation:

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.

(44) PisEmptyProof (EQUIVALENT transformation)

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

(45) TRUE

(46) Obligation:

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

U(ok(X1), ok(X2), ok(X3), ok(X4)) → U(X1, X2, X3, X4)
U(mark(X1), X2, X3, 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.

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U(ok(X1), ok(X2), ok(X3), ok(X4)) → U(X1, X2, X3, X4)

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

Recursive Path Order [RPO].
Precedence:

active₁ > [fst₁, u₂, proper₁] > [natsFrom₁, cons₂] > [U₁, ok₁, s₁, head₁] > nil
active₁ > [fst₁, u₂, proper₁] > pair₂ > [U₁, ok₁, s₁, head₁] > nil
active₁ > [fst₁, u₂, proper₁] > 0 > nil
active₁ > [fst₁, u₂, proper₁] > afterNth₁ > [U₁, ok₁, s₁, head₁] > nil
active₁ > [fst₁, u₂, proper₁] > take₂ > [U₁, ok₁, s₁, head₁] > nil
top > [fst₁, u₂, proper₁] > [natsFrom₁, cons₂] > [U₁, ok₁, s₁, head₁] > nil
top > [fst₁, u₂, proper₁] > pair₂ > [U₁, ok₁, s₁, head₁] > nil
top > [fst₁, u₂, proper₁] > 0 > nil
top > [fst₁, u₂, proper₁] > afterNth₁ > [U₁, ok₁, s₁, head₁] > nil
top > [fst₁, u₂, proper₁] > take₂ > [U₁, ok₁, s₁, head₁] > nil

The following usable rules [FROCOS05] were oriented:

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))

(48) Obligation:

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

U(mark(X1), X2, X3, 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.

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Recursive Path Order [RPO].
Precedence:

[active₁, splitAt₂, sel₂, take₂, top] > s₁ > [cons₂, fst₁, snd₁, tail₁, ok] > natsFrom₁ > mark₁
[active₁, splitAt₂, sel₂, take₂, top] > s₁ > [cons₂, fst₁, snd₁, tail₁, ok] > u₃ > mark₁
[active₁, splitAt₂, sel₂, take₂, top] > pair₂ > [cons₂, fst₁, snd₁, tail₁, ok] > natsFrom₁ > mark₁
[active₁, splitAt₂, sel₂, take₂, top] > pair₂ > [cons₂, fst₁, snd₁, tail₁, ok] > u₃ > mark₁
[active₁, splitAt₂, sel₂, take₂, top] > afterNth₂ > [cons₂, fst₁, snd₁, tail₁, ok] > natsFrom₁ > mark₁
[active₁, splitAt₂, sel₂, take₂, top] > afterNth₂ > [cons₂, fst₁, snd₁, tail₁, ok] > u₃ > mark₁
0 > pair₂ > [cons₂, fst₁, snd₁, tail₁, ok] > natsFrom₁ > mark₁
0 > pair₂ > [cons₂, fst₁, snd₁, tail₁, ok] > u₃ > mark₁
0 > nil > [cons₂, fst₁, snd₁, tail₁, ok] > natsFrom₁ > mark₁
0 > nil > [cons₂, fst₁, snd₁, tail₁, ok] > u₃ > mark₁

The following usable rules [FROCOS05] were oriented:

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))

(50) Obligation:

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.

(51) PisEmptyProof (EQUIVALENT transformation)

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

(52) TRUE

(53) Obligation:

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

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

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Recursive Path Order [RPO].
Precedence:

[active₁, natsFrom₁, proper₁] > pair₂ > [cons₂, u₄] > mark₁
[active₁, natsFrom₁, proper₁] > splitAt₂ > [cons₂, u₄] > mark₁
[active₁, natsFrom₁, proper₁] > 0 > nil > mark₁
[active₁, natsFrom₁, proper₁] > sel₂ > afterNth₂ > mark₁
[active₁, natsFrom₁, proper₁] > take₂ > fst₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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))

(55) Obligation:

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

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.

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SPLITAT(ok(X1), ok(X2)) → SPLITAT(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

[proper, top] > [SPLITAT₁, mark, ok₁, 0, nil]

The following usable rules [FROCOS05] were oriented:

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))

(57) Obligation:

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

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

The TRS R consists of the following rules:

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Recursive Path Order [RPO].
Precedence:

SPLITAT₁ > [mark₁, fst₁, tail₁]
0 > [mark₁, fst₁, tail₁]
top > active₁ > natsFrom₁ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > pair₂ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > nil > [mark₁, fst₁, tail₁]
top > active₁ > u₂ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > sel₂ > [mark₁, fst₁, tail₁]
top > active₁ > afterNth₂ > snd₁ > [mark₁, fst₁, tail₁]
top > active₁ > afterNth₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]

The following usable rules [FROCOS05] were oriented:

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))

(59) Obligation:

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.

(60) PisEmptyProof (EQUIVALENT transformation)

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

(61) TRUE

(62) Obligation:

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

SND(ok(X)) → SND(X)
SND(mark(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.

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SND(mark(X)) → SND(X)

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

Recursive Path Order [RPO].
Precedence:

SND₁ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > cons₂ > [splitAt₂, u₄] > pair₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > 0 > pair₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > nil > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > sel₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > afterNth₂ > snd₁ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > afterNth₂ > [splitAt₂, u₄] > pair₂ > [mark₁, natsFrom₁, fst₁]
top > [mark₁, natsFrom₁, fst₁]

The following usable rules [FROCOS05] were oriented:

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))

(64) Obligation:

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

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.

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SND(ok(X)) → SND(X)

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

Recursive Path Order [RPO].
Precedence:

[active₁, cons₂, nil, afterNth₂, take₁, proper₁] > [SND₁, ok₁, fst₁, splitAt₁, head₁] > top > mark
[active₁, cons₂, nil, afterNth₂, take₁, proper₁] > 0 > mark

The following usable rules [FROCOS05] were oriented:

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))

(66) Obligation:

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.

(67) PisEmptyProof (EQUIVALENT transformation)

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

(68) TRUE

(69) Obligation:

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

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

(70) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(X1, mark(X2)) → PAIR(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

[active₁, natsFrom₁, proper₁] > pair₂ > [cons₂, u₄] > mark₁
[active₁, natsFrom₁, proper₁] > splitAt₂ > [cons₂, u₄] > mark₁
[active₁, natsFrom₁, proper₁] > 0 > nil > mark₁
[active₁, natsFrom₁, proper₁] > sel₂ > afterNth₂ > mark₁
[active₁, natsFrom₁, proper₁] > take₂ > fst₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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))

(71) Obligation:

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

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

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(ok(X1), ok(X2)) → PAIR(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

[proper, top] > [PAIR₁, mark, ok₁, 0, nil]

The following usable rules [FROCOS05] were oriented:

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))

(73) Obligation:

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

PAIR(mark(X1), X2) → PAIR(X1, X2)

The TRS R consists of the following rules:

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(mark(X1), X2) → PAIR(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

PAIR₁ > [mark₁, fst₁, tail₁]
0 > [mark₁, fst₁, tail₁]
top > active₁ > natsFrom₁ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > pair₂ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > nil > [mark₁, fst₁, tail₁]
top > active₁ > u₂ > cons₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]
top > active₁ > sel₂ > [mark₁, fst₁, tail₁]
top > active₁ > afterNth₂ > snd₁ > [mark₁, fst₁, tail₁]
top > active₁ > afterNth₂ > [splitAt₂, take₂] > [mark₁, fst₁, tail₁]

The following usable rules [FROCOS05] were oriented:

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))

(75) Obligation:

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.

(76) PisEmptyProof (EQUIVALENT transformation)

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

(77) TRUE

(78) Obligation:

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.

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FST(mark(X)) → FST(X)

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

Recursive Path Order [RPO].
Precedence:

FST₁ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > cons₂ > [splitAt₂, u₄] > pair₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > 0 > pair₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > nil > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > sel₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > afterNth₂ > snd₁ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > afterNth₂ > [splitAt₂, u₄] > pair₂ > [mark₁, natsFrom₁, fst₁]
top > [mark₁, natsFrom₁, fst₁]

The following usable rules [FROCOS05] were oriented:

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))

(80) Obligation:

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

FST(ok(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.

(81) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FST(ok(X)) → FST(X)

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

Recursive Path Order [RPO].
Precedence:

[active₁, cons₂, nil, afterNth₂, take₁, proper₁] > [FST₁, ok₁, fst₁, splitAt₁, head₁] > top > mark
[active₁, cons₂, nil, afterNth₂, take₁, proper₁] > 0 > mark

The following usable rules [FROCOS05] were oriented:

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))

(82) Obligation:

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.

(83) PisEmptyProof (EQUIVALENT transformation)

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

(84) TRUE

(85) Obligation:

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.

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

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

Recursive Path Order [RPO].
Precedence:

S₁ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > cons₂ > [splitAt₂, u₄] > pair₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > 0 > pair₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > nil > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > sel₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > afterNth₂ > snd₁ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > afterNth₂ > [splitAt₂, u₄] > pair₂ > [mark₁, natsFrom₁, fst₁]
top > [mark₁, natsFrom₁, fst₁]

The following usable rules [FROCOS05] were oriented:

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))

(87) Obligation:

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

S(ok(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.

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(ok(X)) → S(X)

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

Recursive Path Order [RPO].
Precedence:

[active₁, cons₂, nil, afterNth₂, take₁, proper₁] > [S₁, ok₁, fst₁, splitAt₁, head₁] > top > mark
[active₁, cons₂, nil, afterNth₂, take₁, proper₁] > 0 > mark

The following usable rules [FROCOS05] were oriented:

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))

(89) Obligation:

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.

(90) PisEmptyProof (EQUIVALENT transformation)

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

(91) TRUE

(92) Obligation:

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

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

(93) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

proper₁ > [active₁, pair₂, u₃, sel₂, afterNth₂] > cons₂ > splitAt₂ > mark₁
proper₁ > [active₁, pair₂, u₃, sel₂, afterNth₂] > s₁ > splitAt₂ > mark₁
proper₁ > [active₁, pair₂, u₃, sel₂, afterNth₂] > [0, nil] > mark₁
proper₁ > [active₁, pair₂, u₃, sel₂, afterNth₂] > take₂ > splitAt₂ > mark₁

The following usable rules [FROCOS05] were oriented:

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))

(94) Obligation:

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

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.

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(ok(X1), ok(X2)) → CONS(X1, X2)

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

Recursive Path Order [RPO].
Precedence:

CONS₁ > [ok₁, head₁, tail₁]
0 > [cons₂, pair₂] > [ok₁, head₁, tail₁]
0 > nil > [ok₁, head₁, tail₁]
top > proper₁ > [active₁, snd₁, splitAt₂, u₃, afterNth₂] > [cons₂, pair₂] > [ok₁, head₁, tail₁]
top > proper₁ > [active₁, snd₁, splitAt₂, u₃, afterNth₂] > sel₂ > [ok₁, head₁, tail₁]
top > proper₁ > [active₁, snd₁, splitAt₂, u₃, afterNth₂] > take₂ > [ok₁, head₁, tail₁]

The following usable rules [FROCOS05] were oriented:

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))

(96) Obligation:

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.

(97) PisEmptyProof (EQUIVALENT transformation)

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

(98) TRUE

(99) Obligation:

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.

(100) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NATSFROM(mark(X)) → NATSFROM(X)

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

Recursive Path Order [RPO].
Precedence:

NATSFROM₁ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > cons₂ > [splitAt₂, u₄] > pair₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > 0 > pair₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > nil > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > sel₂ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > afterNth₂ > snd₁ > [mark₁, natsFrom₁, fst₁]
active₁ > [take₂, proper₁] > afterNth₂ > [splitAt₂, u₄] > pair₂ > [mark₁, natsFrom₁, fst₁]
top > [mark₁, natsFrom₁, fst₁]

The following usable rules [FROCOS05] were oriented:

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))

(101) Obligation:

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

NATSFROM(ok(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.

(102) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NATSFROM(ok(X)) → NATSFROM(X)

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

Recursive Path Order [RPO].
Precedence:

[active₁, cons₂, nil, afterNth₂, take₁, proper₁] > [NATSFROM₁, ok₁, fst₁, splitAt₁, head₁] > top > mark
[active₁, cons₂, nil, afterNth₂, take₁, proper₁] > 0 > mark

The following usable rules [FROCOS05] were oriented:

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))

(103) Obligation:

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.

(104) PisEmptyProof (EQUIVALENT transformation)

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

(105) TRUE

(106) Obligation:

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

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

(107) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(pair(X1, X2)) → PROPER(X1)
PROPER(pair(X1, X2)) → PROPER(X2)
PROPER(splitAt(X1, X2)) → PROPER(X1)
PROPER(splitAt(X1, X2)) → PROPER(X2)
PROPER(u(X1, X2, X3, X4)) → PROPER(X1)
PROPER(u(X1, X2, X3, X4)) → PROPER(X2)
PROPER(u(X1, X2, X3, X4)) → PROPER(X3)
PROPER(u(X1, X2, X3, X4)) → PROPER(X4)
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(sel(X1, X2)) → PROPER(X2)
PROPER(afterNth(X1, X2)) → PROPER(X1)
PROPER(afterNth(X1, X2)) → PROPER(X2)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)

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

Recursive Path Order [RPO].
Precedence:

[afterNth₂, active₁] > pair₂ > cons₂ > [splitAt₂, u₄, take₂, 0, nil] > [mark₁, top]
[afterNth₂, active₁] > sel₂ > [mark₁, top]

The following usable rules [FROCOS05] were oriented:

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))

(108) Obligation:

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

PROPER(natsFrom(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(fst(X)) → PROPER(X)
PROPER(snd(X)) → PROPER(X)
PROPER(head(X)) → PROPER(X)
PROPER(tail(X)) → PROPER(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.

(109) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(natsFrom(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)

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

Recursive Path Order [RPO].
Precedence:

active₁ > [cons₂, u₂] > [pair₂, splitAt₂] > [mark, top] > [natsFrom₁, s₁] > PROPER₁ > take₂
active₁ > [cons₂, u₂] > [pair₂, splitAt₂] > [mark, top] > afterNth₂ > take₂
active₁ > nil > take₂
active₁ > sel₂ > [mark, top] > [natsFrom₁, s₁] > PROPER₁ > take₂
active₁ > sel₂ > [mark, top] > afterNth₂ > take₂
proper₁ > [cons₂, u₂] > [pair₂, splitAt₂] > [mark, top] > [natsFrom₁, s₁] > PROPER₁ > take₂
proper₁ > [cons₂, u₂] > [pair₂, splitAt₂] > [mark, top] > afterNth₂ > take₂
proper₁ > 0 > take₂
proper₁ > sel₂ > [mark, top] > [natsFrom₁, s₁] > PROPER₁ > take₂
proper₁ > sel₂ > [mark, top] > afterNth₂ > take₂

The following usable rules [FROCOS05] were oriented:

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))

(110) Obligation:

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

PROPER(fst(X)) → PROPER(X)
PROPER(snd(X)) → PROPER(X)
PROPER(head(X)) → PROPER(X)
PROPER(tail(X)) → PROPER(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.

(111) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(fst(X)) → PROPER(X)
PROPER(head(X)) → PROPER(X)
PROPER(tail(X)) → PROPER(X)

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

Recursive Path Order [RPO].
Precedence:

cons > [proper₁, top] > [fst₁, take] > splitAt > mark
cons > [proper₁, top] > head₁ > mark
cons > [proper₁, top] > tail₁ > mark
cons > [proper₁, top] > natsFrom > s > splitAt > mark
cons > [proper₁, top] > afterNth > splitAt > mark
0 > nil > mark
sel > [proper₁, top] > [fst₁, take] > splitAt > mark
sel > [proper₁, top] > head₁ > mark
sel > [proper₁, top] > tail₁ > mark
sel > [proper₁, top] > natsFrom > s > splitAt > mark
sel > [proper₁, top] > afterNth > splitAt > mark

The following usable rules [FROCOS05] were oriented:

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))

(112) Obligation:

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

PROPER(snd(X)) → PROPER(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.

(113) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(snd(X)) → PROPER(X)

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

Recursive Path Order [RPO].
Precedence:

[cons₂, pair₂, u₄, take₂, proper₁] > snd₁ > [natsFrom₁, mark, 0]
[cons₂, pair₂, u₄, take₂, proper₁] > s₁ > [natsFrom₁, mark, 0]
[cons₂, pair₂, u₄, take₂, proper₁] > fst₁ > [natsFrom₁, mark, 0]
[cons₂, pair₂, u₄, take₂, proper₁] > nil > [natsFrom₁, mark, 0]
[cons₂, pair₂, u₄, take₂, proper₁] > sel > [natsFrom₁, mark, 0]
top > [natsFrom₁, mark, 0]

The following usable rules [FROCOS05] were oriented:

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))

(114) Obligation:

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.

(115) PisEmptyProof (EQUIVALENT transformation)

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

(116) TRUE

(117) Obligation:

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

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

(118) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(pair(X1, X2)) → ACTIVE(X1)
ACTIVE(pair(X1, X2)) → ACTIVE(X2)
ACTIVE(splitAt(X1, X2)) → ACTIVE(X1)
ACTIVE(splitAt(X1, X2)) → ACTIVE(X2)
ACTIVE(u(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X2)
ACTIVE(afterNth(X1, X2)) → ACTIVE(X1)
ACTIVE(afterNth(X1, X2)) → ACTIVE(X2)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)

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

Recursive Path Order [RPO].
Precedence:

active₁ > [u₁, sel₂, take₂, mark, proper₁] > cons₂ > [ok, top]
active₁ > [u₁, sel₂, take₂, mark, proper₁] > [splitAt₂, afterNth₂] > pair₂ > [ok, top]
active₁ > [u₁, sel₂, take₂, mark, proper₁] > 0
nil > [ok, top]

The following usable rules [FROCOS05] were oriented:

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))

(119) Obligation:

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

ACTIVE(natsFrom(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(fst(X)) → ACTIVE(X)
ACTIVE(snd(X)) → ACTIVE(X)
ACTIVE(head(X)) → ACTIVE(X)
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.

(120) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(natsFrom(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)

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

Recursive Path Order [RPO].
Precedence:

active₁ > [cons₂, u₂] > [pair₂, splitAt₂] > [mark, top] > [natsFrom₁, s₁] > ACTIVE₁ > take₂
active₁ > [cons₂, u₂] > [pair₂, splitAt₂] > [mark, top] > afterNth₂ > take₂
active₁ > nil > take₂
active₁ > sel₂ > [mark, top] > [natsFrom₁, s₁] > ACTIVE₁ > take₂
active₁ > sel₂ > [mark, top] > afterNth₂ > take₂
proper₁ > [cons₂, u₂] > [pair₂, splitAt₂] > [mark, top] > [natsFrom₁, s₁] > ACTIVE₁ > take₂
proper₁ > [cons₂, u₂] > [pair₂, splitAt₂] > [mark, top] > afterNth₂ > take₂
proper₁ > 0 > take₂
proper₁ > sel₂ > [mark, top] > [natsFrom₁, s₁] > ACTIVE₁ > take₂
proper₁ > sel₂ > [mark, top] > afterNth₂ > take₂

The following usable rules [FROCOS05] were oriented:

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))

(121) Obligation:

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

ACTIVE(fst(X)) → ACTIVE(X)
ACTIVE(snd(X)) → ACTIVE(X)
ACTIVE(head(X)) → ACTIVE(X)
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.

(122) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(fst(X)) → ACTIVE(X)
ACTIVE(head(X)) → ACTIVE(X)
ACTIVE(tail(X)) → ACTIVE(X)

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

Recursive Path Order [RPO].
Precedence:

cons > [proper₁, top] > [fst₁, take] > splitAt > mark
cons > [proper₁, top] > head₁ > mark
cons > [proper₁, top] > tail₁ > mark
cons > [proper₁, top] > natsFrom > s > splitAt > mark
cons > [proper₁, top] > afterNth > splitAt > mark
0 > nil > mark
sel > [proper₁, top] > [fst₁, take] > splitAt > mark
sel > [proper₁, top] > head₁ > mark
sel > [proper₁, top] > tail₁ > mark
sel > [proper₁, top] > natsFrom > s > splitAt > mark
sel > [proper₁, top] > afterNth > splitAt > mark

The following usable rules [FROCOS05] were oriented:

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))

(123) Obligation:

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

ACTIVE(snd(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.

(124) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(snd(X)) → ACTIVE(X)

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

Recursive Path Order [RPO].
Precedence:

[cons₂, pair₂, u₄, take₂, proper₁] > snd₁ > [natsFrom₁, mark, 0]
[cons₂, pair₂, u₄, take₂, proper₁] > s₁ > [natsFrom₁, mark, 0]
[cons₂, pair₂, u₄, take₂, proper₁] > fst₁ > [natsFrom₁, mark, 0]
[cons₂, pair₂, u₄, take₂, proper₁] > nil > [natsFrom₁, mark, 0]
[cons₂, pair₂, u₄, take₂, proper₁] > sel > [natsFrom₁, mark, 0]
top > [natsFrom₁, mark, 0]

The following usable rules [FROCOS05] were oriented:

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))

(125) Obligation:

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.

(126) PisEmptyProof (EQUIVALENT transformation)

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

(127) TRUE

(128) Obligation:

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

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(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.