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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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(U101(tt, N, XS)) → FST(splitAt(N, XS))
ACTIVE(U101(tt, N, XS)) → SPLITAT(N, XS)
ACTIVE(U11(tt, N, XS)) → SND(splitAt(N, XS))
ACTIVE(U11(tt, N, XS)) → SPLITAT(N, XS)
ACTIVE(U41(tt, N)) → CONS(N, natsFrom(s(N)))
ACTIVE(U41(tt, N)) → NATSFROM(s(N))
ACTIVE(U41(tt, N)) → S(N)
ACTIVE(U51(tt, N, XS)) → HEAD(afterNth(N, XS))
ACTIVE(U51(tt, N, XS)) → AFTERNTH(N, XS)
ACTIVE(U71(tt, XS)) → PAIR(nil, XS)
ACTIVE(U81(tt, N, X, XS)) → U82¹(splitAt(N, XS), X)
ACTIVE(U81(tt, N, X, XS)) → SPLITAT(N, XS)
ACTIVE(U82(pair(YS, ZS), X)) → PAIR(cons(X, YS), ZS)
ACTIVE(U82(pair(YS, ZS), X)) → CONS(X, YS)
ACTIVE(afterNth(N, XS)) → U11¹(and(isNatural(N), isLNat(XS)), N, XS)
ACTIVE(afterNth(N, XS)) → AND(isNatural(N), isLNat(XS))
ACTIVE(afterNth(N, XS)) → ISNATURAL(N)
ACTIVE(afterNth(N, XS)) → ISLNAT(XS)
ACTIVE(fst(pair(X, Y))) → U21¹(and(isLNat(X), isLNat(Y)), X)
ACTIVE(fst(pair(X, Y))) → AND(isLNat(X), isLNat(Y))
ACTIVE(fst(pair(X, Y))) → ISLNAT(X)
ACTIVE(fst(pair(X, Y))) → ISLNAT(Y)
ACTIVE(head(cons(N, XS))) → U31¹(and(isNatural(N), isLNat(XS)), N)
ACTIVE(head(cons(N, XS))) → AND(isNatural(N), isLNat(XS))
ACTIVE(head(cons(N, XS))) → ISNATURAL(N)
ACTIVE(head(cons(N, XS))) → ISLNAT(XS)
ACTIVE(isLNat(afterNth(V1, V2))) → AND(isNatural(V1), isLNat(V2))
ACTIVE(isLNat(afterNth(V1, V2))) → ISNATURAL(V1)
ACTIVE(isLNat(afterNth(V1, V2))) → ISLNAT(V2)
ACTIVE(isLNat(cons(V1, V2))) → AND(isNatural(V1), isLNat(V2))
ACTIVE(isLNat(cons(V1, V2))) → ISNATURAL(V1)
ACTIVE(isLNat(cons(V1, V2))) → ISLNAT(V2)
ACTIVE(isLNat(fst(V1))) → ISPLNAT(V1)
ACTIVE(isLNat(natsFrom(V1))) → ISNATURAL(V1)
ACTIVE(isLNat(snd(V1))) → ISPLNAT(V1)
ACTIVE(isLNat(tail(V1))) → ISLNAT(V1)
ACTIVE(isLNat(take(V1, V2))) → AND(isNatural(V1), isLNat(V2))
ACTIVE(isLNat(take(V1, V2))) → ISNATURAL(V1)
ACTIVE(isLNat(take(V1, V2))) → ISLNAT(V2)
ACTIVE(isNatural(head(V1))) → ISLNAT(V1)
ACTIVE(isNatural(s(V1))) → ISNATURAL(V1)
ACTIVE(isNatural(sel(V1, V2))) → AND(isNatural(V1), isLNat(V2))
ACTIVE(isNatural(sel(V1, V2))) → ISNATURAL(V1)
ACTIVE(isNatural(sel(V1, V2))) → ISLNAT(V2)
ACTIVE(isPLNat(pair(V1, V2))) → AND(isLNat(V1), isLNat(V2))
ACTIVE(isPLNat(pair(V1, V2))) → ISLNAT(V1)
ACTIVE(isPLNat(pair(V1, V2))) → ISLNAT(V2)
ACTIVE(isPLNat(splitAt(V1, V2))) → AND(isNatural(V1), isLNat(V2))
ACTIVE(isPLNat(splitAt(V1, V2))) → ISNATURAL(V1)
ACTIVE(isPLNat(splitAt(V1, V2))) → ISLNAT(V2)
ACTIVE(natsFrom(N)) → U41¹(isNatural(N), N)
ACTIVE(natsFrom(N)) → ISNATURAL(N)
ACTIVE(sel(N, XS)) → U51¹(and(isNatural(N), isLNat(XS)), N, XS)
ACTIVE(sel(N, XS)) → AND(isNatural(N), isLNat(XS))
ACTIVE(sel(N, XS)) → ISNATURAL(N)
ACTIVE(sel(N, XS)) → ISLNAT(XS)
ACTIVE(snd(pair(X, Y))) → U61¹(and(isLNat(X), isLNat(Y)), Y)
ACTIVE(snd(pair(X, Y))) → AND(isLNat(X), isLNat(Y))
ACTIVE(snd(pair(X, Y))) → ISLNAT(X)
ACTIVE(snd(pair(X, Y))) → ISLNAT(Y)
ACTIVE(splitAt(0, XS)) → U71¹(isLNat(XS), XS)
ACTIVE(splitAt(0, XS)) → ISLNAT(XS)
ACTIVE(splitAt(s(N), cons(X, XS))) → U81¹(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)
ACTIVE(splitAt(s(N), cons(X, XS))) → AND(isNatural(N), and(isNatural(X), isLNat(XS)))
ACTIVE(splitAt(s(N), cons(X, XS))) → ISNATURAL(N)
ACTIVE(splitAt(s(N), cons(X, XS))) → AND(isNatural(X), isLNat(XS))
ACTIVE(splitAt(s(N), cons(X, XS))) → ISNATURAL(X)
ACTIVE(splitAt(s(N), cons(X, XS))) → ISLNAT(XS)
ACTIVE(tail(cons(N, XS))) → U91¹(and(isNatural(N), isLNat(XS)), XS)
ACTIVE(tail(cons(N, XS))) → AND(isNatural(N), isLNat(XS))
ACTIVE(tail(cons(N, XS))) → ISNATURAL(N)
ACTIVE(tail(cons(N, XS))) → ISLNAT(XS)
ACTIVE(take(N, XS)) → U101¹(and(isNatural(N), isLNat(XS)), N, XS)
ACTIVE(take(N, XS)) → AND(isNatural(N), isLNat(XS))
ACTIVE(take(N, XS)) → ISNATURAL(N)
ACTIVE(take(N, XS)) → ISLNAT(XS)
ACTIVE(U101(X1, X2, X3)) → U101¹(active(X1), X2, X3)
ACTIVE(U101(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(fst(X)) → FST(active(X))
ACTIVE(fst(X)) → ACTIVE(X)
ACTIVE(splitAt(X1, X2)) → SPLITAT(active(X1), X2)
ACTIVE(splitAt(X1, X2)) → ACTIVE(X1)
ACTIVE(splitAt(X1, X2)) → SPLITAT(X1, active(X2))
ACTIVE(splitAt(X1, X2)) → ACTIVE(X2)
ACTIVE(U11(X1, X2, X3)) → U11¹(active(X1), X2, X3)
ACTIVE(U11(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(snd(X)) → SND(active(X))
ACTIVE(snd(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → U21¹(active(X1), X2)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U31(X1, X2)) → U31¹(active(X1), X2)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U41(X1, X2)) → U41¹(active(X1), X2)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(natsFrom(X)) → NATSFROM(active(X))
ACTIVE(natsFrom(X)) → ACTIVE(X)
ACTIVE(s(X)) → S(active(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2, X3)) → U51¹(active(X1), X2, X3)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(head(X)) → HEAD(active(X))
ACTIVE(head(X)) → ACTIVE(X)
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(U61(X1, X2)) → U61¹(active(X1), X2)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U71(X1, X2)) → U71¹(active(X1), X2)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
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(U81(X1, X2, X3, X4)) → U81¹(active(X1), X2, X3, X4)
ACTIVE(U81(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U82(X1, X2)) → U82¹(active(X1), X2)
ACTIVE(U82(X1, X2)) → ACTIVE(X1)
ACTIVE(U91(X1, X2)) → U91¹(active(X1), X2)
ACTIVE(U91(X1, X2)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → AND(active(X1), X2)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
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)
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)
U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
FST(mark(X)) → FST(X)
SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)
U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
SND(mark(X)) → SND(X)
U21¹(mark(X1), X2) → U21¹(X1, X2)
U31¹(mark(X1), X2) → U31¹(X1, X2)
U41¹(mark(X1), X2) → U41¹(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
NATSFROM(mark(X)) → NATSFROM(X)
S(mark(X)) → S(X)
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
HEAD(mark(X)) → HEAD(X)
AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
U61¹(mark(X1), X2) → U61¹(X1, X2)
U71¹(mark(X1), X2) → U71¹(X1, X2)
PAIR(mark(X1), X2) → PAIR(X1, X2)
PAIR(X1, mark(X2)) → PAIR(X1, X2)
U81¹(mark(X1), X2, X3, X4) → U81¹(X1, X2, X3, X4)
U82¹(mark(X1), X2) → U82¹(X1, X2)
U91¹(mark(X1), X2) → U91¹(X1, X2)
AND(mark(X1), X2) → AND(X1, X2)
TAIL(mark(X)) → TAIL(X)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(X1, mark(X2)) → SEL(X1, X2)
PROPER(U101(X1, X2, X3)) → U101¹(proper(X1), proper(X2), proper(X3))
PROPER(U101(X1, X2, X3)) → PROPER(X1)
PROPER(U101(X1, X2, X3)) → PROPER(X2)
PROPER(U101(X1, X2, X3)) → PROPER(X3)
PROPER(fst(X)) → FST(proper(X))
PROPER(fst(X)) → PROPER(X)
PROPER(splitAt(X1, X2)) → SPLITAT(proper(X1), proper(X2))
PROPER(splitAt(X1, X2)) → PROPER(X1)
PROPER(splitAt(X1, X2)) → PROPER(X2)
PROPER(U11(X1, X2, X3)) → U11¹(proper(X1), proper(X2), proper(X3))
PROPER(U11(X1, X2, X3)) → PROPER(X1)
PROPER(U11(X1, X2, X3)) → PROPER(X2)
PROPER(U11(X1, X2, X3)) → PROPER(X3)
PROPER(snd(X)) → SND(proper(X))
PROPER(snd(X)) → PROPER(X)
PROPER(U21(X1, X2)) → U21¹(proper(X1), proper(X2))
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U31(X1, X2)) → U31¹(proper(X1), proper(X2))
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2)) → U41¹(proper(X1), proper(X2))
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U41(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(natsFrom(X)) → NATSFROM(proper(X))
PROPER(natsFrom(X)) → PROPER(X)
PROPER(s(X)) → S(proper(X))
PROPER(s(X)) → PROPER(X)
PROPER(U51(X1, X2, X3)) → U51¹(proper(X1), proper(X2), proper(X3))
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(head(X)) → HEAD(proper(X))
PROPER(head(X)) → PROPER(X)
PROPER(afterNth(X1, X2)) → AFTERNTH(proper(X1), proper(X2))
PROPER(afterNth(X1, X2)) → PROPER(X1)
PROPER(afterNth(X1, X2)) → PROPER(X2)
PROPER(U61(X1, X2)) → U61¹(proper(X1), proper(X2))
PROPER(U61(X1, X2)) → PROPER(X1)
PROPER(U61(X1, X2)) → PROPER(X2)
PROPER(U71(X1, X2)) → U71¹(proper(X1), proper(X2))
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U71(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(U81(X1, X2, X3, X4)) → U81¹(proper(X1), proper(X2), proper(X3), proper(X4))
PROPER(U81(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U81(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U81(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U81(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U82(X1, X2)) → U82¹(proper(X1), proper(X2))
PROPER(U82(X1, X2)) → PROPER(X1)
PROPER(U82(X1, X2)) → PROPER(X2)
PROPER(U91(X1, X2)) → U91¹(proper(X1), proper(X2))
PROPER(U91(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → AND(proper(X1), proper(X2))
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNatural(X)) → ISNATURAL(proper(X))
PROPER(isNatural(X)) → PROPER(X)
PROPER(isLNat(X)) → ISLNAT(proper(X))
PROPER(isLNat(X)) → PROPER(X)
PROPER(isPLNat(X)) → ISPLNAT(proper(X))
PROPER(isPLNat(X)) → PROPER(X)
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)
PROPER(sel(X1, X2)) → SEL(proper(X1), proper(X2))
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(sel(X1, X2)) → PROPER(X2)
U101¹(ok(X1), ok(X2), ok(X3)) → U101¹(X1, X2, X3)
FST(ok(X)) → FST(X)
SPLITAT(ok(X1), ok(X2)) → SPLITAT(X1, X2)
U11¹(ok(X1), ok(X2), ok(X3)) → U11¹(X1, X2, X3)
SND(ok(X)) → SND(X)
U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)
U31¹(ok(X1), ok(X2)) → U31¹(X1, X2)
U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
NATSFROM(ok(X)) → NATSFROM(X)
S(ok(X)) → S(X)
U51¹(ok(X1), ok(X2), ok(X3)) → U51¹(X1, X2, X3)
HEAD(ok(X)) → HEAD(X)
AFTERNTH(ok(X1), ok(X2)) → AFTERNTH(X1, X2)
U61¹(ok(X1), ok(X2)) → U61¹(X1, X2)
U71¹(ok(X1), ok(X2)) → U71¹(X1, X2)
PAIR(ok(X1), ok(X2)) → PAIR(X1, X2)
U81¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U81¹(X1, X2, X3, X4)
U82¹(ok(X1), ok(X2)) → U82¹(X1, X2)
U91¹(ok(X1), ok(X2)) → U91¹(X1, X2)
AND(ok(X1), ok(X2)) → AND(X1, X2)
ISNATURAL(ok(X)) → ISNATURAL(X)
ISLNAT(ok(X)) → ISLNAT(X)
ISPLNAT(ok(X)) → ISPLNAT(X)
TAIL(ok(X)) → TAIL(X)
TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
SEL(ok(X1), ok(X2)) → SEL(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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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 30 SCCs with 134 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

ISPLNAT(ok(X)) → ISPLNAT(X)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

ISPLNAT(ok(X)) → ISPLNAT(X)

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

ok₁ > ISPLNAT₁

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

ISLNAT(ok(X)) → ISLNAT(X)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISLNAT(ok(X)) → ISLNAT(X)

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

ok₁ > ISLNAT₁

The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) Obligation:

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

ISNATURAL(ok(X)) → ISNATURAL(X)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATURAL(ok(X)) → ISNATURAL(X)

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

ok₁ > ISNATURAL₁

The following usable rules [FROCOS05] were oriented: none

(17) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(18) PisEmptyProof (EQUIVALENT transformation)

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

(19) TRUE

(20) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(21) 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)
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(x1)
ok(x1)  =  ok(x1)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > SEL₁

The following usable rules [FROCOS05] were oriented: none

(22) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(mark(X1), X2) → SEL(X1, X2)

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

Lexicographic Path Order [LPO].
Precedence:

mark₁ > SEL₁

The following usable rules [FROCOS05] were oriented: none

(24) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(25) PisEmptyProof (EQUIVALENT transformation)

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

(26) TRUE

(27) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(X1, mark(X2)) → TAKE(X1, X2)
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(x1)
ok(x1)  =  ok(x1)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > TAKE₁

The following usable rules [FROCOS05] were oriented: none

(29) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(mark(X1), X2) → TAKE(X1, X2)

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

Lexicographic Path Order [LPO].
Precedence:

mark₁ > TAKE₁

The following usable rules [FROCOS05] were oriented: none

(31) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(32) PisEmptyProof (EQUIVALENT transformation)

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

(33) TRUE

(34) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(ok(X)) → TAIL(X)

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

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(36) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

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

mark₁ > TAIL₁

The following usable rules [FROCOS05] were oriented: none

(38) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(39) PisEmptyProof (EQUIVALENT transformation)

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

(40) TRUE

(41) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(mark(X1), 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)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(43) Obligation:

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

AND(ok(X1), ok(X2)) → AND(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(44) 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)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > AND₁

The following usable rules [FROCOS05] were oriented: none

(45) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(46) PisEmptyProof (EQUIVALENT transformation)

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

(47) TRUE

(48) Obligation:

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

U91¹(ok(X1), ok(X2)) → U91¹(X1, X2)
U91¹(mark(X1), X2) → U91¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(mark(X1), X2) → U91¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(50) Obligation:

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

U91¹(ok(X1), ok(X2)) → U91¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(ok(X1), ok(X2)) → U91¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2)  =  U91¹(x2)
ok(x1)  =  ok(x1)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > U91^1₁

The following usable rules [FROCOS05] were oriented: none

(52) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(53) PisEmptyProof (EQUIVALENT transformation)

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

(54) TRUE

(55) Obligation:

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

U82¹(ok(X1), ok(X2)) → U82¹(X1, X2)
U82¹(mark(X1), X2) → U82¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(mark(X1), X2) → U82¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(57) Obligation:

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

U82¹(ok(X1), ok(X2)) → U82¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(ok(X1), ok(X2)) → U82¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2)  =  U82¹(x2)
ok(x1)  =  ok(x1)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > U82^1₁

The following usable rules [FROCOS05] were oriented: none

(59) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(60) PisEmptyProof (EQUIVALENT transformation)

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

(61) TRUE

(62) Obligation:

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

U81¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U81¹(X1, X2, X3, X4)
U81¹(mark(X1), X2, X3, X4) → U81¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U81¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3, x4)  =  U81¹(x4)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > U81^1₁
mark₁ > U81^1₁

The following usable rules [FROCOS05] were oriented: none

(64) Obligation:

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

U81¹(mark(X1), X2, X3, X4) → U81¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(mark(X1), X2, X3, X4) → U81¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3, x4)  =  U81¹(x1)
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(66) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(67) PisEmptyProof (EQUIVALENT transformation)

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

(68) TRUE

(69) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(70) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(X1, mark(X2)) → PAIR(X1, X2)
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(x1)
ok(x1)  =  ok(x1)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > PAIR₁

The following usable rules [FROCOS05] were oriented: none

(71) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(mark(X1), X2) → PAIR(X1, X2)

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

Lexicographic Path Order [LPO].
Precedence:

mark₁ > PAIR₁

The following usable rules [FROCOS05] were oriented: none

(73) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(74) PisEmptyProof (EQUIVALENT transformation)

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

(75) TRUE

(76) Obligation:

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

U71¹(ok(X1), ok(X2)) → U71¹(X1, X2)
U71¹(mark(X1), X2) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

U71¹(mark(X1), X2) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(78) Obligation:

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

U71¹(ok(X1), ok(X2)) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(ok(X1), ok(X2)) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2)  =  U71¹(x2)
ok(x1)  =  ok(x1)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > U71^1₁

The following usable rules [FROCOS05] were oriented: none

(80) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(81) PisEmptyProof (EQUIVALENT transformation)

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

(82) TRUE

(83) Obligation:

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

U61¹(ok(X1), ok(X2)) → U61¹(X1, X2)
U61¹(mark(X1), X2) → U61¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

U61¹(mark(X1), X2) → U61¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(85) Obligation:

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

U61¹(ok(X1), ok(X2)) → U61¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(ok(X1), ok(X2)) → U61¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2)  =  U61¹(x2)
ok(x1)  =  ok(x1)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > U61^1₁

The following usable rules [FROCOS05] were oriented: none

(87) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(88) PisEmptyProof (EQUIVALENT transformation)

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

(89) TRUE

(90) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
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(x1)
ok(x1)  =  ok(x1)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > AFTERNTH₁

The following usable rules [FROCOS05] were oriented: none

(92) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

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

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

Lexicographic Path Order [LPO].
Precedence:

mark₁ > AFTERNTH₁

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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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:

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

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

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)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(99) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

HEAD(mark(X)) → HEAD(X)

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

mark₁ > HEAD₁

The following usable rules [FROCOS05] were oriented: none

(101) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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) PisEmptyProof (EQUIVALENT transformation)

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

(103) TRUE

(104) Obligation:

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

U51¹(ok(X1), ok(X2), ok(X3)) → U51¹(X1, X2, X3)
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(105) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(ok(X1), ok(X2), ok(X3)) → U51¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2, x3)  =  U51¹(x3)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U51^1₁

The following usable rules [FROCOS05] were oriented: none

(106) Obligation:

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

U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2, x3)  =  U51¹(x1)
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U51^1₁

The following usable rules [FROCOS05] were oriented: none

(108) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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) PisEmptyProof (EQUIVALENT transformation)

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

(110) TRUE

(111) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(112) 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)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(113) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

S(mark(X)) → S(X)

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

mark₁ > S₁

The following usable rules [FROCOS05] were oriented: none

(115) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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) PisEmptyProof (EQUIVALENT transformation)

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

(117) TRUE

(118) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(119) 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)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(120) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

NATSFROM(mark(X)) → NATSFROM(X)

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

mark₁ > NATSFROM₁

The following usable rules [FROCOS05] were oriented: none

(122) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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) PisEmptyProof (EQUIVALENT transformation)

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

(124) TRUE

(125) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(126) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

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

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(127) Obligation:

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

CONS(ok(X1), ok(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(128) 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)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > CONS₁

The following usable rules [FROCOS05] were oriented: none

(129) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(130) PisEmptyProof (EQUIVALENT transformation)

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

(131) TRUE

(132) Obligation:

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

U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)
U41¹(mark(X1), X2) → U41¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(mark(X1), X2) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(134) Obligation:

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

U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(135) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2)  =  U41¹(x2)
ok(x1)  =  ok(x1)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > U41^1₁

The following usable rules [FROCOS05] were oriented: none

(136) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(137) PisEmptyProof (EQUIVALENT transformation)

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

(138) TRUE

(139) Obligation:

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

U31¹(ok(X1), ok(X2)) → U31¹(X1, X2)
U31¹(mark(X1), X2) → U31¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

U31¹(mark(X1), X2) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(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:

U31¹(ok(X1), ok(X2)) → U31¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

U31¹(ok(X1), ok(X2)) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2)  =  U31¹(x2)
ok(x1)  =  ok(x1)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > U31^1₁

The following usable rules [FROCOS05] were oriented: none

(143) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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) PisEmptyProof (EQUIVALENT transformation)

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

(145) TRUE

(146) Obligation:

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

U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(147) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(mark(X1), X2) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(148) Obligation:

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

U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(149) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  U21¹(x2)
ok(x1)  =  ok(x1)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > U21^1₁

The following usable rules [FROCOS05] were oriented: none

(150) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(151) PisEmptyProof (EQUIVALENT transformation)

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

(152) TRUE

(153) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(154) 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)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(155) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

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

mark₁ > SND₁

The following usable rules [FROCOS05] were oriented: none

(157) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(158) PisEmptyProof (EQUIVALENT transformation)

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

(159) TRUE

(160) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(161) 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)) → U11¹(X1, X2, X3)

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

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U11^1₁

The following usable rules [FROCOS05] were oriented: none

(162) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(163) 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) → U11¹(X1, X2, X3)

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

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U11^1₁

The following usable rules [FROCOS05] were oriented: none

(164) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(165) PisEmptyProof (EQUIVALENT transformation)

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

(166) TRUE

(167) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(168) 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)
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(x1)
ok(x1)  =  ok(x1)

Lexicographic Path Order [LPO].
Precedence:

ok₁ > SPLITAT₁

The following usable rules [FROCOS05] were oriented: none

(169) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(170) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic Path Order [LPO].
Precedence:

mark₁ > SPLITAT₁

The following usable rules [FROCOS05] were oriented: none

(171) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(172) PisEmptyProof (EQUIVALENT transformation)

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

(173) TRUE

(174) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(175) 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)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(176) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

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

mark₁ > FST₁

The following usable rules [FROCOS05] were oriented: none

(178) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(179) PisEmptyProof (EQUIVALENT transformation)

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

(180) TRUE

(181) Obligation:

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

U101¹(ok(X1), ok(X2), ok(X3)) → U101¹(X1, X2, X3)
U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(182) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(ok(X1), ok(X2), ok(X3)) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  U101¹(x3)
ok(x1)  =  ok(x1)
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U101^1₁

The following usable rules [FROCOS05] were oriented: none

(183) Obligation:

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

U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(184) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  U101¹(x1)
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U101^1₁

The following usable rules [FROCOS05] were oriented: none

(185) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(186) PisEmptyProof (EQUIVALENT transformation)

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

(187) TRUE

(188) Obligation:

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

PROPER(U101(X1, X2, X3)) → PROPER(X2)
PROPER(U101(X1, X2, X3)) → PROPER(X1)
PROPER(U101(X1, X2, X3)) → PROPER(X3)
PROPER(fst(X)) → PROPER(X)
PROPER(splitAt(X1, X2)) → PROPER(X1)
PROPER(splitAt(X1, X2)) → PROPER(X2)
PROPER(U11(X1, X2, X3)) → PROPER(X1)
PROPER(U11(X1, X2, X3)) → PROPER(X2)
PROPER(U11(X1, X2, X3)) → PROPER(X3)
PROPER(snd(X)) → PROPER(X)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(natsFrom(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(head(X)) → PROPER(X)
PROPER(afterNth(X1, X2)) → PROPER(X1)
PROPER(afterNth(X1, X2)) → PROPER(X2)
PROPER(U61(X1, X2)) → PROPER(X1)
PROPER(U61(X1, X2)) → PROPER(X2)
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(pair(X1, X2)) → PROPER(X1)
PROPER(pair(X1, X2)) → PROPER(X2)
PROPER(U81(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U81(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U81(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U81(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U82(X1, X2)) → PROPER(X1)
PROPER(U82(X1, X2)) → PROPER(X2)
PROPER(U91(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNatural(X)) → PROPER(X)
PROPER(isLNat(X)) → PROPER(X)
PROPER(isPLNat(X)) → PROPER(X)
PROPER(tail(X)) → PROPER(X)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(sel(X1, X2)) → PROPER(X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(189) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U101(X1, X2, X3)) → PROPER(X2)
PROPER(U101(X1, X2, X3)) → PROPER(X1)
PROPER(U101(X1, X2, X3)) → PROPER(X3)
PROPER(fst(X)) → PROPER(X)
PROPER(splitAt(X1, X2)) → PROPER(X1)
PROPER(splitAt(X1, X2)) → PROPER(X2)
PROPER(U11(X1, X2, X3)) → PROPER(X1)
PROPER(U11(X1, X2, X3)) → PROPER(X2)
PROPER(U11(X1, X2, X3)) → PROPER(X3)
PROPER(snd(X)) → PROPER(X)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(natsFrom(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(head(X)) → PROPER(X)
PROPER(afterNth(X1, X2)) → PROPER(X1)
PROPER(afterNth(X1, X2)) → PROPER(X2)
PROPER(U61(X1, X2)) → PROPER(X1)
PROPER(U61(X1, X2)) → PROPER(X2)
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(pair(X1, X2)) → PROPER(X1)
PROPER(pair(X1, X2)) → PROPER(X2)
PROPER(U81(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U81(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U81(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U81(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U82(X1, X2)) → PROPER(X1)
PROPER(U82(X1, X2)) → PROPER(X2)
PROPER(U91(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNatural(X)) → PROPER(X)
PROPER(isLNat(X)) → PROPER(X)
PROPER(isPLNat(X)) → PROPER(X)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(sel(X1, X2)) → PROPER(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
U101(x1, x2, x3)  =  U101(x1, x2, x3)
fst(x1)  =  fst(x1)
splitAt(x1, x2)  =  splitAt(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
snd(x1)  =  snd(x1)
U21(x1, x2)  =  U21(x1, x2)
U31(x1, x2)  =  U31(x1, x2)
U41(x1, x2)  =  U41(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
natsFrom(x1)  =  natsFrom(x1)
s(x1)  =  s(x1)
U51(x1, x2, x3)  =  U51(x1, x2, x3)
head(x1)  =  head(x1)
afterNth(x1, x2)  =  afterNth(x1, x2)
U61(x1, x2)  =  U61(x1, x2)
U71(x1, x2)  =  U71(x1, x2)
pair(x1, x2)  =  pair(x1, x2)
U81(x1, x2, x3, x4)  =  U81(x1, x2, x3, x4)
U82(x1, x2)  =  U82(x1, x2)
U91(x1, x2)  =  U91(x1, x2)
and(x1, x2)  =  and(x1, x2)
isNatural(x1)  =  isNatural(x1)
isLNat(x1)  =  isLNat(x1)
isPLNat(x1)  =  isPLNat(x1)
tail(x1)  =  x1
take(x1, x2)  =  take(x1, x2)
sel(x1, x2)  =  sel(x1, x2)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(190) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

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

tail₁ > PROPER₁

The following usable rules [FROCOS05] were oriented: none

(192) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(193) PisEmptyProof (EQUIVALENT transformation)

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

(194) TRUE

(195) Obligation:

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

ACTIVE(fst(X)) → ACTIVE(X)
ACTIVE(U101(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(splitAt(X1, X2)) → ACTIVE(X1)
ACTIVE(splitAt(X1, X2)) → ACTIVE(X2)
ACTIVE(U11(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(snd(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(natsFrom(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(head(X)) → ACTIVE(X)
ACTIVE(afterNth(X1, X2)) → ACTIVE(X1)
ACTIVE(afterNth(X1, X2)) → ACTIVE(X2)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(pair(X1, X2)) → ACTIVE(X1)
ACTIVE(pair(X1, X2)) → ACTIVE(X2)
ACTIVE(U81(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U82(X1, X2)) → ACTIVE(X1)
ACTIVE(U91(X1, X2)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(tail(X)) → ACTIVE(X)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X2)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(196) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(fst(X)) → ACTIVE(X)
ACTIVE(U101(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(splitAt(X1, X2)) → ACTIVE(X1)
ACTIVE(splitAt(X1, X2)) → ACTIVE(X2)
ACTIVE(U11(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(snd(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(natsFrom(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(head(X)) → ACTIVE(X)
ACTIVE(afterNth(X1, X2)) → ACTIVE(X1)
ACTIVE(afterNth(X1, X2)) → ACTIVE(X2)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(pair(X1, X2)) → ACTIVE(X1)
ACTIVE(pair(X1, X2)) → ACTIVE(X2)
ACTIVE(U81(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U82(X1, X2)) → ACTIVE(X1)
ACTIVE(U91(X1, X2)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
fst(x1)  =  fst(x1)
U101(x1, x2, x3)  =  U101(x1)
splitAt(x1, x2)  =  splitAt(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
snd(x1)  =  snd(x1)
U21(x1, x2)  =  U21(x1)
U31(x1, x2)  =  U31(x1)
U41(x1, x2)  =  U41(x1)
cons(x1, x2)  =  cons(x1)
natsFrom(x1)  =  natsFrom(x1)
s(x1)  =  s(x1)
U51(x1, x2, x3)  =  U51(x1)
head(x1)  =  head(x1)
afterNth(x1, x2)  =  afterNth(x1, x2)
U61(x1, x2)  =  U61(x1)
U71(x1, x2)  =  U71(x1)
pair(x1, x2)  =  pair(x1, x2)
U81(x1, x2, x3, x4)  =  U81(x1)
U82(x1, x2)  =  U82(x1)
U91(x1, x2)  =  U91(x1)
and(x1, x2)  =  and(x1)
tail(x1)  =  x1
take(x1, x2)  =  take(x1, x2)
sel(x1, x2)  =  sel(x1, x2)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(197) Obligation:

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

ACTIVE(tail(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(198) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(tail(X)) → ACTIVE(X)

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

tail₁ > ACTIVE₁

The following usable rules [FROCOS05] were oriented: none

(199) Obligation:

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

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.

(200) PisEmptyProof (EQUIVALENT transformation)

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

(201) TRUE

(202) 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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(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.