Termination w.r.t. Q proof of /tmp/tmp2K4PAG/Ex4_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 [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

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

      ↳ QDPOrderProof

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

MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(zWquot(X1, X2)) → MARK(X2)
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(quot(X1, X2)) → MARK(X2)
A__QUOT(s(X), s(Y)) → A__QUOT(a__minus(mark(X), mark(Y)), s(mark(Y)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
MARK(minus(X1, X2)) → A__MINUS(mark(X1), mark(X2))
MARK(zWquot(X1, X2)) → MARK(X1)
MARK(zWquot(X1, X2)) → A__ZWQUOT(mark(X1), mark(X2))
MARK(minus(X1, X2)) → MARK(X2)
A__MINUS(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(Y)
A__QUOT(s(X), s(Y)) → MARK(Y)
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
A__MINUS(s(X), s(Y)) → MARK(Y)
MARK(quot(X1, X2)) → MARK(X1)
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → A__QUOT(mark(X), mark(Y))
A__FROM(X) → MARK(X)
MARK(quot(X1, X2)) → A__QUOT(mark(X1), mark(X2))
A__MINUS(s(X), s(Y)) → MARK(X)
A__QUOT(s(X), s(Y)) → A__MINUS(mark(X), mark(Y))
A__ZWQUOT(cons(X, XS), cons(Y, YS)) → MARK(X)
MARK(minus(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
A__QUOT(s(X), s(Y)) → 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 [15].

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

Used ordering: Polynomial interpretation [25,35]:

POL(sel(x₁, x₂)) = 0
POL(from(x₁)) = 0
POL(minus(x₁, x₂)) = 0
POL(A__ZWQUOT(x₁, x₂)) = x_1
POL(A__MINUS(x₁, x₂)) = x_1
POL(mark(x₁)) = 1/4
POL(a__from(x₁)) = 1/4
POL(a__minus(x₁, x₂)) = (1/4)x_1 + (1/2)x_2
POL(A__SEL(x₁, x₂)) = 1/4
POL(0) = 0
POL(a__zWquot(x₁, x₂)) = x_1
POL(A__QUOT(x₁, x₂)) = x_1
POL(cons(x₁, x₂)) = 1/4
POL(MARK(x₁)) = 1/4
POL(A__FROM(x₁)) = 1/4
POL(a__sel(x₁, x₂)) = x_2
POL(quot(x₁, x₂)) = 0
POL(a__quot(x₁, x₂)) = x_1
POL(s(x₁)) = 1/4
POL(zWquot(x₁, x₂)) = 0
POL(nil) = 0

The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

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

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