Termination w.r.t. Q proof of /tmp/tmpWsX44E/ExSec4_2_DLMMU04

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 QDPOrderProof (⇔)

    ↳9 QDP

    ↳10 PisEmptyProof (⇔)

    ↳11 TRUE

  ↳12 QDP

    ↳13 QDPOrderProof (⇔)

    ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 PisEmptyProof (⇔)

    ↳18 TRUE

  ↳19 QDP

    ↳20 QDPOrderProof (⇔)

    ↳21 QDP

    ↳22 QDPOrderProof (⇔)

    ↳23 QDP

    ↳24 PisEmptyProof (⇔)

    ↳25 TRUE

  ↳26 QDP

    ↳27 QDPOrderProof (⇔)

    ↳28 QDP

    ↳29 QDPOrderProof (⇔)

    ↳30 QDP

    ↳31 PisEmptyProof (⇔)

    ↳32 TRUE

  ↳33 QDP

    ↳34 QDPOrderProof (⇔)

    ↳35 QDP

    ↳36 QDPOrderProof (⇔)

    ↳37 QDP

    ↳38 PisEmptyProof (⇔)

    ↳39 TRUE

  ↳40 QDP

    ↳41 QDPOrderProof (⇔)

    ↳42 QDP

    ↳43 QDPOrderProof (⇔)

    ↳44 QDP

    ↳45 QDPOrderProof (⇔)

    ↳46 QDP

    ↳47 QDPOrderProof (⇔)

    ↳48 QDP

    ↳49 QDPOrderProof (⇔)

    ↳50 QDP

    ↳51 QDPOrderProof (⇔)

    ↳52 QDP

    ↳53 PisEmptyProof (⇔)

    ↳54 TRUE

  ↳55 QDP

    ↳56 QDPOrderProof (⇔)

    ↳57 QDP

    ↳58 QDPOrderProof (⇔)

    ↳59 QDP

    ↳60 PisEmptyProof (⇔)

    ↳61 TRUE

  ↳62 QDP

    ↳63 QDPOrderProof (⇔)

    ↳64 QDP

    ↳65 QDPOrderProof (⇔)

    ↳66 QDP

    ↳67 PisEmptyProof (⇔)

    ↳68 TRUE

  ↳69 QDP

    ↳70 QDPOrderProof (⇔)

    ↳71 QDP

    ↳72 QDPOrderProof (⇔)

    ↳73 QDP

    ↳74 PisEmptyProof (⇔)

    ↳75 TRUE

  ↳76 QDP

    ↳77 QDPOrderProof (⇔)

    ↳78 QDP

    ↳79 QDPOrderProof (⇔)

    ↳80 QDP

    ↳81 PisEmptyProof (⇔)

    ↳82 TRUE

  ↳83 QDP

    ↳84 QDPOrderProof (⇔)

    ↳85 QDP

    ↳86 QDPOrderProof (⇔)

    ↳87 QDP

    ↳88 PisEmptyProof (⇔)

    ↳89 TRUE

  ↳90 QDP

    ↳91 QDPOrderProof (⇔)

    ↳92 QDP

    ↳93 QDPOrderProof (⇔)

    ↳94 QDP

    ↳95 PisEmptyProof (⇔)

    ↳96 TRUE

  ↳97 QDP

    ↳98 QDPOrderProof (⇔)

    ↳99 QDP

    ↳100 QDPOrderProof (⇔)

    ↳101 QDP

    ↳102 PisEmptyProof (⇔)

    ↳103 TRUE

  ↳104 QDP

    ↳105 QDPOrderProof (⇔)

    ↳106 QDP

    ↳107 QDPOrderProof (⇔)

    ↳108 QDP

    ↳109 QDPOrderProof (⇔)

    ↳110 QDP

(0) Obligation:

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

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

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 14 SCCs with 29 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

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

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(X1, mark(X2)) → TAKE(X1, X2)
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, x2)
mark(x1) = mark(x1)
active(x1) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[TAKE₂, mark₁]

Status:

TAKE₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

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

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[TAKE₂, active₁]

Status:

active₁: [1]
TAKE₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

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

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

(10) PisEmptyProof (EQUIVALENT transformation)

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

(11) TRUE

(12) 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(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)

The TRS R consists of the following rules:

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

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

(13) 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(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, x2)
mark(x1) = mark(x1)
active(x1) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[AFTERNTH₂, mark₁]

Status:

mark₁: [1]
AFTERNTH₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)

The TRS R consists of the following rules:

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

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[AFTERNTH₂, active₁]

Status:

active₁: [1]
AFTERNTH₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

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

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

(17) PisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) 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(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)

The TRS R consists of the following rules:

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

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

(20) 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(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, x2)
mark(x1) = mark(x1)
active(x1) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[SEL₂, mark₁]

Status:

mark₁: [1]
SEL₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(21) Obligation:

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

SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)

The TRS R consists of the following rules:

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

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

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[SEL₂, active₁]

Status:

active₁: [1]
SEL₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(23) Obligation:

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

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

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

(24) PisEmptyProof (EQUIVALENT transformation)

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

(25) TRUE

(26) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(active(X)) → TAIL(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[TAIL₁, active₁]

Status:

active₁: [1]
TAIL₁: [1]

The following usable rules [FROCOS05] were oriented: none

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

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

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(mark(X)) → TAIL(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(30) Obligation:

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

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

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

(31) PisEmptyProof (EQUIVALENT transformation)

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

(32) TRUE

(33) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(active(X)) → HEAD(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[HEAD₁, active₁]

Status:

active₁: [1]
HEAD₁: [1]

The following usable rules [FROCOS05] were oriented: none

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

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

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(mark(X)) → HEAD(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(37) Obligation:

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

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

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

(38) PisEmptyProof (EQUIVALENT transformation)

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

(39) TRUE

(40) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(41) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(42) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(43) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

active₁: [1]
U₂: [2,1]
mark: []

The following usable rules [FROCOS05] were oriented: none

(44) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > U₁

Status:

active₁: [1]
U₁: [1]

The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

active: []
U₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(48) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U₂, active₁]

Status:

active₁: [1]
U₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(50) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

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

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

(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:

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

The TRS R consists of the following rules:

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

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

(56) 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(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, x2)
mark(x1) = mark(x1)
active(x1) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[SPLITAT₂, mark₁]

Status:

SPLITAT₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(57) Obligation:

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

SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)

The TRS R consists of the following rules:

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

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[SPLITAT₂, active₁]

Status:

active₁: [1]
SPLITAT₂: [2,1]

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

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

(60) PisEmptyProof (EQUIVALENT transformation)

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

(61) TRUE

(62) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SND(active(X)) → SND(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[SND₁, active₁]

Status:

active₁: [1]
SND₁: [1]

The following usable rules [FROCOS05] were oriented: none

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

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SND(mark(X)) → SND(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

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

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

(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(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)

The TRS R consists of the following rules:

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

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

(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(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, x2)
mark(x1) = mark(x1)
active(x1) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[PAIR₂, mark₁]

Status:

PAIR₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(71) Obligation:

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

PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)

The TRS R consists of the following rules:

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

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

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[PAIR₂, active₁]

Status:

active₁: [1]
PAIR₂: [2,1]

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

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

(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:

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

The TRS R consists of the following rules:

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

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

(77) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FST(active(X)) → FST(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[FST₁, active₁]

Status:

active₁: [1]
FST₁: [1]

The following usable rules [FROCOS05] were oriented: none

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

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

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FST(mark(X)) → FST(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

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

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

(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:

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

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

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

(84) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[S₁, active₁]

Status:

active₁: [1]
S₁: [1]

The following usable rules [FROCOS05] were oriented: none

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

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

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

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

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

(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:

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

The TRS R consists of the following rules:

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

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

(91) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[CONS₂, mark₁]

Status:

CONS₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(92) Obligation:

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

CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

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

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

(93) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[CONS₂, active₁]

Status:

active₁: [1]
CONS₂: [2,1]

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

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

(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:

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

The TRS R consists of the following rules:

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

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

(98) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NATSFROM(active(X)) → NATSFROM(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[NATSFROM₁, active₁]

Status:

active₁: [1]
NATSFROM₁: [1]

The following usable rules [FROCOS05] were oriented: none

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

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

(100) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NATSFROM(mark(X)) → NATSFROM(X)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

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

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

(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:

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

The TRS R consists of the following rules:

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

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

(105) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(pair(X1, X2)) → ACTIVE(pair(mark(X1), mark(X2)))

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

0 > mark > [MARK, natsFrom, cons, s, fst, snd, splitAt, u, head, tail, sel, afterNth, take] > pair
0 > mark > nil > pair

Status:

MARK: []
sel: []
fst: []
s: []
natsFrom: []
0: []
afterNth: []
take: []
snd: []
splitAt: []
tail: []
cons: []
u: []
head: []
mark: []
nil: []
pair: []

The following usable rules [FROCOS05] were oriented:

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

(106) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(107) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(s(X)) → ACTIVE(s(mark(X)))

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[mark₁, active₁] > pair > [MARK, natsFrom, cons, fst, snd, splitAt, u, head, tail, sel, afterNth, take] > s
[mark₁, active₁] > 0
[mark₁, active₁] > nil

Status:

MARK: []
sel: []
mark₁: [1]
fst: []
s: []
natsFrom: []
0: []
afterNth: []
take: []
snd: []
splitAt: []
tail: []
active₁: [1]
cons: []
u: []
head: []
nil: []
pair: []

The following usable rules [FROCOS05] were oriented:

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

(108) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(109) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[MARK, natsFrom, mark, fst, snd, splitAt, u, head, tail, sel, afterNth, take] > cons > [s, active] > pair
[MARK, natsFrom, mark, fst, snd, splitAt, u, head, tail, sel, afterNth, take] > 0 > [s, active] > pair
[MARK, natsFrom, mark, fst, snd, splitAt, u, head, tail, sel, afterNth, take] > nil > [s, active] > pair

Status:

active: []
MARK: []
sel: []
fst: []
s: []
natsFrom: []
0: []
afterNth: []
take: []
snd: []
splitAt: []
tail: []
cons: []
u: []
head: []
mark: []
nil: []
pair: []

The following usable rules [FROCOS05] were oriented:

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

(110) Obligation:

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

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

The TRS R consists of the following rules:

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

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