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

(0) Obligation:

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

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
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(U11(tt, N, X, XS)) → U12¹(splitAt(N, XS), X)
ACTIVE(U11(tt, N, X, XS)) → SPLITAT(N, XS)
ACTIVE(U12(pair(YS, ZS), X)) → PAIR(cons(X, YS), ZS)
ACTIVE(U12(pair(YS, ZS), X)) → CONS(X, YS)
ACTIVE(afterNth(N, XS)) → SND(splitAt(N, XS))
ACTIVE(afterNth(N, XS)) → SPLITAT(N, XS)
ACTIVE(natsFrom(N)) → CONS(N, natsFrom(s(N)))
ACTIVE(natsFrom(N)) → NATSFROM(s(N))
ACTIVE(natsFrom(N)) → S(N)
ACTIVE(sel(N, XS)) → HEAD(afterNth(N, XS))
ACTIVE(sel(N, XS)) → AFTERNTH(N, XS)
ACTIVE(splitAt(0, XS)) → PAIR(nil, XS)
ACTIVE(splitAt(s(N), cons(X, XS))) → U11¹(tt, N, X, XS)
ACTIVE(take(N, XS)) → FST(splitAt(N, XS))
ACTIVE(take(N, XS)) → SPLITAT(N, XS)
ACTIVE(U11(X1, X2, X3, X4)) → U11¹(active(X1), X2, X3, X4)
ACTIVE(U11(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U12(X1, X2)) → U12¹(active(X1), X2)
ACTIVE(U12(X1, X2)) → ACTIVE(X1)
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(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(cons(X1, X2)) → CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
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(snd(X)) → SND(active(X))
ACTIVE(snd(X)) → ACTIVE(X)
ACTIVE(and(X1, X2)) → AND(active(X1), X2)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(fst(X)) → FST(active(X))
ACTIVE(fst(X)) → ACTIVE(X)
ACTIVE(head(X)) → HEAD(active(X))
ACTIVE(head(X)) → ACTIVE(X)
ACTIVE(natsFrom(X)) → NATSFROM(active(X))
ACTIVE(natsFrom(X)) → ACTIVE(X)
ACTIVE(s(X)) → S(active(X))
ACTIVE(s(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(tail(X)) → TAIL(active(X))
ACTIVE(tail(X)) → ACTIVE(X)
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)
U11¹(mark(X1), X2, X3, X4) → U11¹(X1, X2, X3, X4)
U12¹(mark(X1), X2) → U12¹(X1, X2)
SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)
PAIR(mark(X1), X2) → PAIR(X1, X2)
PAIR(X1, mark(X2)) → PAIR(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
SND(mark(X)) → SND(X)
AND(mark(X1), X2) → AND(X1, X2)
FST(mark(X)) → FST(X)
HEAD(mark(X)) → HEAD(X)
NATSFROM(mark(X)) → NATSFROM(X)
S(mark(X)) → S(X)
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(X1, mark(X2)) → SEL(X1, X2)
TAIL(mark(X)) → TAIL(X)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
PROPER(U11(X1, X2, X3, X4)) → U11¹(proper(X1), proper(X2), proper(X3), proper(X4))
PROPER(U11(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U11(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U11(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U11(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U12(X1, X2)) → U12¹(proper(X1), proper(X2))
PROPER(U12(X1, X2)) → PROPER(X1)
PROPER(U12(X1, X2)) → PROPER(X2)
PROPER(splitAt(X1, X2)) → SPLITAT(proper(X1), proper(X2))
PROPER(splitAt(X1, X2)) → PROPER(X1)
PROPER(splitAt(X1, X2)) → PROPER(X2)
PROPER(pair(X1, X2)) → PAIR(proper(X1), proper(X2))
PROPER(pair(X1, X2)) → PROPER(X1)
PROPER(pair(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(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(snd(X)) → SND(proper(X))
PROPER(snd(X)) → PROPER(X)
PROPER(and(X1, X2)) → AND(proper(X1), proper(X2))
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(fst(X)) → FST(proper(X))
PROPER(fst(X)) → PROPER(X)
PROPER(head(X)) → HEAD(proper(X))
PROPER(head(X)) → PROPER(X)
PROPER(natsFrom(X)) → NATSFROM(proper(X))
PROPER(natsFrom(X)) → PROPER(X)
PROPER(s(X)) → S(proper(X))
PROPER(s(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(tail(X)) → TAIL(proper(X))
PROPER(tail(X)) → PROPER(X)
PROPER(take(X1, X2)) → TAKE(proper(X1), proper(X2))
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
U11¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U11¹(X1, X2, X3, X4)
U12¹(ok(X1), ok(X2)) → U12¹(X1, X2)
SPLITAT(ok(X1), ok(X2)) → SPLITAT(X1, X2)
PAIR(ok(X1), ok(X2)) → PAIR(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
AFTERNTH(ok(X1), ok(X2)) → AFTERNTH(X1, X2)
SND(ok(X)) → SND(X)
AND(ok(X1), ok(X2)) → AND(X1, X2)
FST(ok(X)) → FST(X)
HEAD(ok(X)) → HEAD(X)
NATSFROM(ok(X)) → NATSFROM(X)
S(ok(X)) → S(X)
SEL(ok(X1), ok(X2)) → SEL(X1, X2)
TAIL(ok(X)) → TAIL(X)
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(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
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 18 SCCs with 52 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(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
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)  =  TAKE(x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

mark₁ > TAKE₁

The following usable rules [FROCOS05] were oriented: none

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

Lexicographic Path Order [LPO].
Precedence:

ok₁ > [TAKE₁, mark]

The following usable rules [FROCOS05] were oriented: none

(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(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
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: Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

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

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

TAIL(ok(X)) → TAIL(X)
TAIL(mark(X)) → TAIL(X)

The TRS R consists of the following rules:

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

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)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

[TAIL₁, ok₁]

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

TAIL(mark(X)) → TAIL(X)

The TRS R consists of the following rules:

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

TAIL(mark(X)) → TAIL(X)

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

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

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

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

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

(20) TRUE

(21) Obligation:

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

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

The TRS R consists of the following rules:

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

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(X1, mark(X2)) → SEL(X1, X2)

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(x1)
ok(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

mark₁ > SEL₁

The following usable rules [FROCOS05] were oriented: none

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

Lexicographic Path Order [LPO].
Precedence:

ok₁ > [SEL₁, mark]

The following usable rules [FROCOS05] were oriented: none

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

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

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

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

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

(29) TRUE

(30) Obligation:

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

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

(31) 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)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

[S₁, ok₁]

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

S(mark(X)) → S(X)

The TRS R consists of the following rules:

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

S(mark(X)) → S(X)

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

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(34) Obligation:

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

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

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

(36) TRUE

(37) Obligation:

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

NATSFROM(ok(X)) → NATSFROM(X)
NATSFROM(mark(X)) → NATSFROM(X)

The TRS R consists of the following rules:

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

(38) 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)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

[NATSFROM₁, ok₁]

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

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

NATSFROM(mark(X)) → NATSFROM(X)

The TRS R consists of the following rules:

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

NATSFROM(mark(X)) → NATSFROM(X)

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

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

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

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

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

(43) TRUE

(44) Obligation:

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

HEAD(ok(X)) → HEAD(X)
HEAD(mark(X)) → HEAD(X)

The TRS R consists of the following rules:

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

(45) 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)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

[HEAD₁, ok₁]

The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

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

HEAD(mark(X)) → HEAD(X)

The TRS R consists of the following rules:

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

HEAD(mark(X)) → HEAD(X)

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

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(48) Obligation:

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

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

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

(50) TRUE

(51) Obligation:

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

FST(ok(X)) → FST(X)
FST(mark(X)) → FST(X)

The TRS R consists of the following rules:

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

(52) 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)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

[FST₁, ok₁]

The following usable rules [FROCOS05] were oriented: none

(53) Obligation:

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

FST(mark(X)) → FST(X)

The TRS R consists of the following rules:

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

FST(mark(X)) → FST(X)

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

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(55) Obligation:

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

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

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

(57) TRUE

(58) Obligation:

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

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

The TRS R consists of the following rules:

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

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(ok(X1), ok(X2)) → AND(X1, X2)

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

Lexicographic Path Order [LPO].
Precedence:

ok₁ > AND₁
mark > AND₁

The following usable rules [FROCOS05] were oriented: none

(60) Obligation:

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

AND(mark(X1), X2) → AND(X1, X2)

The TRS R consists of the following rules:

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

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(mark(X1), X2) → AND(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(62) Obligation:

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

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

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

(64) TRUE

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

(66) 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)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

[SND₁, ok₁]

The following usable rules [FROCOS05] were oriented: none

(67) Obligation:

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

SND(mark(X)) → SND(X)

The TRS R consists of the following rules:

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

(68) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SND(mark(X)) → SND(X)

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

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(69) Obligation:

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

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

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

(71) TRUE

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

(73) 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)  =  AFTERNTH(x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

mark₁ > AFTERNTH₁

The following usable rules [FROCOS05] were oriented: none

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

(75) 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)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > [AFTERNTH₁, mark]

The following usable rules [FROCOS05] were oriented: none

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

(77) 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: Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(78) Obligation:

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

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

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

(80) TRUE

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

(82) 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(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Lexicographic Path Order [LPO].
Precedence:

ok₁ > CONS₁
mark > CONS₁

The following usable rules [FROCOS05] were oriented: none

(83) Obligation:

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

CONS(mark(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

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

(84) 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: Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(85) Obligation:

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

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

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

(87) TRUE

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

(89) 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)  =  PAIR(x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

mark₁ > PAIR₁

The following usable rules [FROCOS05] were oriented: none

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

(91) 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)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > [PAIR₁, mark]

The following usable rules [FROCOS05] were oriented: none

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

PAIR(mark(X1), X2) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(94) Obligation:

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

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

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

(96) TRUE

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

(98) 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)  =  SPLITAT(x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

mark₁ > SPLITAT₁

The following usable rules [FROCOS05] were oriented: none

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

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)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > [SPLITAT₁, mark]

The following usable rules [FROCOS05] were oriented: none

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

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

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(103) Obligation:

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

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

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

The TRS R consists of the following rules:

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

U12¹(ok(X1), ok(X2)) → U12¹(X1, X2)

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

Lexicographic Path Order [LPO].
Precedence:

ok₁ > U12^1₁
mark > U12^1₁

The following usable rules [FROCOS05] were oriented: none

(108) Obligation:

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

U12¹(mark(X1), X2) → U12¹(X1, X2)

The TRS R consists of the following rules:

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

U12¹(mark(X1), X2) → U12¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(110) Obligation:

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

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

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

(112) TRUE

(113) Obligation:

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

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

The TRS R consists of the following rules:

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

(114) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic Path Order [LPO].
Precedence:

ok₁ > U11^1₁

The following usable rules [FROCOS05] were oriented: none

(115) Obligation:

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

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

The TRS R consists of the following rules:

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

(116) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(117) Obligation:

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

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

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

(119) TRUE

(120) Obligation:

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

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

The TRS R consists of the following rules:

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

(121) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U11(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U11(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U11(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U11(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U12(X1, X2)) → PROPER(X1)
PROPER(U12(X1, X2)) → PROPER(X2)
PROPER(splitAt(X1, X2)) → PROPER(X1)
PROPER(splitAt(X1, X2)) → PROPER(X2)
PROPER(pair(X1, X2)) → PROPER(X1)
PROPER(pair(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(afterNth(X1, X2)) → PROPER(X1)
PROPER(afterNth(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(sel(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)  =  PROPER(x1)
U11(x1, x2, x3, x4)  =  U11(x1, x2, x3, x4)
U12(x1, x2)  =  U12(x1, x2)
splitAt(x1, x2)  =  splitAt(x1, x2)
pair(x1, x2)  =  pair(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
afterNth(x1, x2)  =  afterNth(x1, x2)
snd(x1)  =  x1
and(x1, x2)  =  and(x1, x2)
fst(x1)  =  x1
head(x1)  =  x1
natsFrom(x1)  =  x1
s(x1)  =  x1
sel(x1, x2)  =  sel(x1, x2)
tail(x1)  =  x1
take(x1, x2)  =  take(x1, x2)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(122) Obligation:

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

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

The TRS R consists of the following rules:

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

(123) 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)  =  PROPER(x1)
snd(x1)  =  snd(x1)
fst(x1)  =  x1
head(x1)  =  x1
natsFrom(x1)  =  x1
s(x1)  =  x1
tail(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(124) Obligation:

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

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

The TRS R consists of the following rules:

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

(125) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
fst(x1)  =  fst(x1)
head(x1)  =  x1
natsFrom(x1)  =  x1
s(x1)  =  x1
tail(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(126) Obligation:

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

PROPER(head(X)) → PROPER(X)
PROPER(natsFrom(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(tail(X)) → PROPER(X)

The TRS R consists of the following rules:

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

(127) 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)
head(x1)  =  x1
natsFrom(x1)  =  natsFrom(x1)
s(x1)  =  s(x1)
tail(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(128) Obligation:

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

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

The TRS R consists of the following rules:

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

(129) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(head(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)
head(x1)  =  head(x1)
tail(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

[PROPER₁, head₁]

The following usable rules [FROCOS05] were oriented: none

(130) Obligation:

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

PROPER(tail(X)) → PROPER(X)

The TRS R consists of the following rules:

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

(131) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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
tail(x1)  =  tail(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(132) Obligation:

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

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

(133) PisEmptyProof (EQUIVALENT transformation)

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

(134) TRUE

(135) Obligation:

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

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

The TRS R consists of the following rules:

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

(136) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(splitAt(X1, X2)) → ACTIVE(X1)
ACTIVE(splitAt(X1, X2)) → ACTIVE(X2)
ACTIVE(pair(X1, X2)) → ACTIVE(X1)
ACTIVE(pair(X1, X2)) → ACTIVE(X2)
ACTIVE(afterNth(X1, X2)) → ACTIVE(X1)
ACTIVE(afterNth(X1, X2)) → ACTIVE(X2)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(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)  =  ACTIVE(x1)
U12(x1, x2)  =  x1
U11(x1, x2, x3, x4)  =  x1
splitAt(x1, x2)  =  splitAt(x1, x2)
pair(x1, x2)  =  pair(x1, x2)
cons(x1, x2)  =  x1
afterNth(x1, x2)  =  afterNth(x1, x2)
snd(x1)  =  x1
and(x1, x2)  =  and(x1)
fst(x1)  =  x1
head(x1)  =  x1
natsFrom(x1)  =  x1
s(x1)  =  x1
sel(x1, x2)  =  sel(x1, x2)
tail(x1)  =  x1
take(x1, x2)  =  take(x1, x2)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(137) Obligation:

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

ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(snd(X)) → ACTIVE(X)
ACTIVE(fst(X)) → ACTIVE(X)
ACTIVE(head(X)) → ACTIVE(X)
ACTIVE(natsFrom(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(tail(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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

(138) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U12(x1, x2)  =  x1
U11(x1, x2, x3, x4)  =  x1
cons(x1, x2)  =  cons(x1)
snd(x1)  =  x1
fst(x1)  =  x1
head(x1)  =  x1
natsFrom(x1)  =  x1
s(x1)  =  x1
tail(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(139) Obligation:

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

ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(snd(X)) → ACTIVE(X)
ACTIVE(fst(X)) → ACTIVE(X)
ACTIVE(head(X)) → ACTIVE(X)
ACTIVE(natsFrom(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(tail(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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

(140) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(snd(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)  =  ACTIVE(x1)
U12(x1, x2)  =  U12(x1)
U11(x1, x2, x3, x4)  =  U11(x1, x2, x3, x4)
snd(x1)  =  snd(x1)
fst(x1)  =  x1
head(x1)  =  x1
natsFrom(x1)  =  x1
s(x1)  =  x1
tail(x1)  =  tail(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(141) Obligation:

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

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

The TRS R consists of the following rules:

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

(142) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(head(X)) → ACTIVE(X)
ACTIVE(natsFrom(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)
fst(x1)  =  x1
head(x1)  =  head(x1)
natsFrom(x1)  =  natsFrom(x1)
s(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(143) Obligation:

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

ACTIVE(fst(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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

(144) 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)

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

Lexicographic Path Order [LPO].
Precedence:

[ACTIVE₁, fst₁]

The following usable rules [FROCOS05] were oriented: none

(145) Obligation:

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

ACTIVE(s(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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

(146) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)  =  x1
s(x1)  =  s(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(147) Obligation:

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

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

(148) PisEmptyProof (EQUIVALENT transformation)

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

(149) TRUE

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