Termination w.r.t. Q proof of TRS/TRCSREx4_7_37_Bor03

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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.

↳ QTRS

  ↳ DependencyPairsProof

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.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

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

MARK(zWquot(X1, X2)) → MARK(X2)
A__SEL(0, cons(X, XS)) → MARK(X)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(Y)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__MINUS(s(X), s(Y)) → MARK(Y)
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(X1, X2)) → MARK(X1)
A__QUOT(s(X), s(Y)) → MARK(Y)
A__QUOT(s(X), s(Y)) → A__QUOT(a__minus(mark(X), mark(Y)), s(mark(Y)))
MARK(quot(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(quot(X1, X2)) → MARK(X2)
MARK(minus(X1, X2)) → MARK(X2)
A__QUOT(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
MARK(minus(X1, X2)) → A__MINUS(mark(X1), mark(X2))
MARK(quot(X1, X2)) → A__QUOT(mark(X1), mark(X2))
MARK(minus(X1, X2)) → MARK(X1)
A__MINUS(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → A__QUOT(mark(X), mark(Y))
MARK(s(X)) → MARK(X)
MARK(zWquot(X1, X2)) → A__ZWQUOT(mark(X1), mark(X2))
MARK(zWquot(X1, X2)) → MARK(X1)
A__QUOT(s(X), s(Y)) → MARK(X)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(X)
A__MINUS(s(X), s(Y)) → MARK(X)
A__FROM(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.
We deleted some edges using various graph approximations

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ QDPOrderProof

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

MARK(zWquot(X1, X2)) → MARK(X2)
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(Y)
A__MINUS(s(X), s(Y)) → MARK(Y)
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(X1, X2)) → MARK(X1)
A__QUOT(s(X), s(Y)) → MARK(Y)
MARK(quot(X1, X2)) → MARK(X1)
A__QUOT(s(X), s(Y)) → A__QUOT(a__minus(mark(X), mark(Y)), s(mark(Y)))
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(cons(X1, X2)) → MARK(X1)
MARK(minus(X1, X2)) → MARK(X2)
MARK(quot(X1, X2)) → MARK(X2)
A__QUOT(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
MARK(minus(X1, X2)) → A__MINUS(mark(X1), mark(X2))
MARK(quot(X1, X2)) → A__QUOT(mark(X1), mark(X2))
MARK(minus(X1, X2)) → MARK(X1)
A__MINUS(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → A__QUOT(mark(X), mark(Y))
MARK(s(X)) → MARK(X)
MARK(zWquot(X1, X2)) → A__ZWQUOT(mark(X1), mark(X2))
MARK(zWquot(X1, X2)) → MARK(X1)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(X)
A__QUOT(s(X), s(Y)) → MARK(X)
A__MINUS(s(X), s(Y)) → MARK(X)
A__FROM(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.
We use the reduction pair processor [13].

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.

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

Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK
zWquot(x1, x2) = zWquot
A__SEL(x1, x2) = A__SEL
0 = 0
cons(x1, x2) = cons
sel(x1, x2) = sel
mark(x1) = mark
A__ZWQUOT(x1, x2) = A__ZWQUOT
A__MINUS(x1, x2) = x2
s(x1) = s
A__QUOT(x1, x2) = x1
quot(x1, x2) = x2
a__minus(x1, x2) = a__minus
from(x1) = from
A__FROM(x1) = A__FROM
minus(x1, x2) = minus
a__quot(x1, x2) = x2
nil = nil
a__zWquot(x1, x2) = x2
a__from(x1) = a__from
a__sel(x1, x2) = a__sel

Lexicographic Path Order [19].
Precedence:

[MARK, A_{_SEL}, cons, mark, A_{_ZWQUOT}, s, A_{_FROM}, a_{_from}, a_{_sel]} > [0, a_{_minus]} > [zWquot, sel, minus, nil]
[MARK, A_{_SEL}, cons, mark, A_{_ZWQUOT}, s, A_{_FROM}, a_{_from}, a_{_sel]} > from > [zWquot, sel, minus, nil]

The following usable rules [14] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

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

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