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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 QDPOrderProof (⇔)
↳4 QDP

(0) Obligation:

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

a__from(X) → cons(mark(X), from(s(X)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__minus(X, 0) → 0
a__minus(s(X), s(Y)) → a__minus(mark(X), mark(Y))
a__quot(0, s(Y)) → 0
a__quot(s(X), s(Y)) → s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
a__zWquot(XS, nil) → nil
a__zWquot(nil, XS) → nil
a__zWquot(cons(X, XS), cons(Y, YS)) → cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
mark(from(X)) → a__from(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(minus(X1, X2)) → a__minus(mark(X1), mark(X2))
mark(quot(X1, X2)) → a__quot(mark(X1), mark(X2))
mark(zWquot(X1, X2)) → a__zWquot(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__sel(X1, X2) → sel(X1, X2)
a__minus(X1, X2) → minus(X1, X2)
a__quot(X1, X2) → quot(X1, X2)
a__zWquot(X1, X2) → zWquot(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:

A__FROM(X) → MARK(X)
A__SEL(0, cons(X, XS)) → MARK(X)
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__MINUS(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
A__MINUS(s(X), s(Y)) → MARK(X)
A__MINUS(s(X), s(Y)) → MARK(Y)
A__QUOT(s(X), s(Y)) → A__QUOT(a__minus(mark(X), mark(Y)), s(mark(Y)))
A__QUOT(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
A__QUOT(s(X), s(Y)) → MARK(X)
A__QUOT(s(X), s(Y)) → MARK(Y)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → A__QUOT(mark(X), mark(Y))
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(X)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(Y)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(minus(X1, X2)) → A__MINUS(mark(X1), mark(X2))
MARK(minus(X1, X2)) → MARK(X1)
MARK(minus(X1, X2)) → MARK(X2)
MARK(quot(X1, X2)) → A__QUOT(mark(X1), mark(X2))
MARK(quot(X1, X2)) → MARK(X1)
MARK(quot(X1, X2)) → MARK(X2)
MARK(zWquot(X1, X2)) → A__ZWQUOT(mark(X1), mark(X2))
MARK(zWquot(X1, X2)) → MARK(X1)
MARK(zWquot(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__minus(X, 0) → 0
a__minus(s(X), s(Y)) → a__minus(mark(X), mark(Y))
a__quot(0, s(Y)) → 0
a__quot(s(X), s(Y)) → s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
a__zWquot(XS, nil) → nil
a__zWquot(nil, XS) → nil
a__zWquot(cons(X, XS), cons(Y, YS)) → cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
mark(from(X)) → a__from(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(minus(X1, X2)) → a__minus(mark(X1), mark(X2))
mark(quot(X1, X2)) → a__quot(mark(X1), mark(X2))
mark(zWquot(X1, X2)) → a__zWquot(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__sel(X1, X2) → sel(X1, X2)
a__minus(X1, X2) → minus(X1, X2)
a__quot(X1, X2) → quot(X1, X2)
a__zWquot(X1, X2) → zWquot(X1, X2)

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

(3) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__QUOT(s(X), s(Y)) → A__QUOT(a__minus(mark(X), mark(Y)), s(mark(Y)))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__FROM(x1)  =  A__FROM
MARK(x1)  =  MARK
A__SEL(x1, x2)  =  A__SEL
0  =  0
cons(x1, x2)  =  cons
s(x1)  =  s
mark(x1)  =  mark
A__MINUS(x1, x2)  =  A__MINUS
A__QUOT(x1, x2)  =  x1
a__minus(x1, x2)  =  a__minus
A__ZWQUOT(x1, x2)  =  x2
from(x1)  =  from
sel(x1, x2)  =  sel
minus(x1, x2)  =  minus
quot(x1, x2)  =  quot
zWquot(x1, x2)  =  x1
a__from(x1)  =  a__from
a__sel(x1, x2)  =  x2
a__quot(x1, x2)  =  a__quot
a__zWquot(x1, x2)  =  x1
nil  =  nil

Lexicographic path order with status [LPO].
Quasi-Precedence:
[AFROM, MARK, ASEL, cons, s, mark, AMINUS, quot, afrom, aquot] > [aminus, minus] > 0 > [sel, nil]
[AFROM, MARK, ASEL, cons, s, mark, AMINUS, quot, afrom, aquot] > from > [sel, nil]

Status:
AFROM: []
MARK: []
ASEL: []
0: []
cons: []
s: []
mark: []
AMINUS: []
aminus: []
from: []
sel: []
minus: []
quot: []
afrom: []
aquot: []
nil: []


The following usable rules [FROCOS05] were oriented:

mark(from(X)) → a__from(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(minus(X1, X2)) → a__minus(mark(X1), mark(X2))
mark(quot(X1, X2)) → a__quot(mark(X1), mark(X2))
mark(zWquot(X1, X2)) → a__zWquot(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__minus(X, 0) → 0
a__minus(s(X), s(Y)) → a__minus(mark(X), mark(Y))
a__minus(X1, X2) → minus(X1, X2)
a__from(X) → from(X)
a__sel(X1, X2) → sel(X1, X2)
a__quot(0, s(Y)) → 0
a__quot(X1, X2) → quot(X1, X2)
a__zWquot(XS, nil) → nil
a__zWquot(nil, XS) → nil
a__zWquot(X1, X2) → zWquot(X1, X2)
a__zWquot(cons(X, XS), cons(Y, YS)) → cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
a__quot(s(X), s(Y)) → s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
a__from(X) → cons(mark(X), from(s(X)))

(4) Obligation:

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

A__FROM(X) → MARK(X)
A__SEL(0, cons(X, XS)) → MARK(X)
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__MINUS(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
A__MINUS(s(X), s(Y)) → MARK(X)
A__MINUS(s(X), s(Y)) → MARK(Y)
A__QUOT(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
A__QUOT(s(X), s(Y)) → MARK(X)
A__QUOT(s(X), s(Y)) → MARK(Y)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → A__QUOT(mark(X), mark(Y))
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(X)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(Y)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(minus(X1, X2)) → A__MINUS(mark(X1), mark(X2))
MARK(minus(X1, X2)) → MARK(X1)
MARK(minus(X1, X2)) → MARK(X2)
MARK(quot(X1, X2)) → A__QUOT(mark(X1), mark(X2))
MARK(quot(X1, X2)) → MARK(X1)
MARK(quot(X1, X2)) → MARK(X2)
MARK(zWquot(X1, X2)) → A__ZWQUOT(mark(X1), mark(X2))
MARK(zWquot(X1, X2)) → MARK(X1)
MARK(zWquot(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__minus(X, 0) → 0
a__minus(s(X), s(Y)) → a__minus(mark(X), mark(Y))
a__quot(0, s(Y)) → 0
a__quot(s(X), s(Y)) → s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
a__zWquot(XS, nil) → nil
a__zWquot(nil, XS) → nil
a__zWquot(cons(X, XS), cons(Y, YS)) → cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
mark(from(X)) → a__from(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(minus(X1, X2)) → a__minus(mark(X1), mark(X2))
mark(quot(X1, X2)) → a__quot(mark(X1), mark(X2))
mark(zWquot(X1, X2)) → a__zWquot(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__sel(X1, X2) → sel(X1, X2)
a__minus(X1, X2) → minus(X1, X2)
a__quot(X1, X2) → quot(X1, X2)
a__zWquot(X1, X2) → zWquot(X1, X2)

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