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__from₁(X) -> cons₂(mark₁(X), from₁(s₁(X)))
a__sel₂(0, cons₂(X, XS)) -> mark₁(X)
a__sel₂(s₁(N), cons₂(X, XS)) -> a__sel₂(mark₁(N), mark₁(XS))
a__minus₂(X, 0) -> 0
a__minus₂(s₁(X), s₁(Y)) -> a__minus₂(mark₁(X), mark₁(Y))
a__quot₂(0, s₁(Y)) -> 0
a__quot₂(s₁(X), s₁(Y)) -> s₁(a__quot₂(a__minus₂(mark₁(X), mark₁(Y)), s₁(mark₁(Y))))
a__zWquot₂(XS, nil) -> nil
a__zWquot₂(nil, XS) -> nil
a__zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(a__quot₂(mark₁(X), mark₁(Y)), zWquot₂(XS, YS))
mark₁(from₁(X)) -> a__from₁(mark₁(X))
mark₁(sel₂(X1, X2)) -> a__sel₂(mark₁(X1), mark₁(X2))
mark₁(minus₂(X1, X2)) -> a__minus₂(mark₁(X1), mark₁(X2))
mark₁(quot₂(X1, X2)) -> a__quot₂(mark₁(X1), mark₁(X2))
mark₁(zWquot₂(X1, X2)) -> a__zWquot₂(mark₁(X1), mark₁(X2))
mark₁(cons₂(X1, X2)) -> cons₂(mark₁(X1), X2)
mark₁(s₁(X)) -> s₁(mark₁(X))
mark₁(0) -> 0
mark₁(nil) -> nil
a__from₁(X) -> from₁(X)
a__sel₂(X1, X2) -> sel₂(X1, X2)
a__minus₂(X1, X2) -> minus₂(X1, X2)
a__quot₂(X1, X2) -> quot₂(X1, X2)
a__zWquot₂(X1, X2) -> zWquot₂(X1, X2)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a__from₁(X) -> cons₂(mark₁(X), from₁(s₁(X)))
a__sel₂(0, cons₂(X, XS)) -> mark₁(X)
a__sel₂(s₁(N), cons₂(X, XS)) -> a__sel₂(mark₁(N), mark₁(XS))
a__minus₂(X, 0) -> 0
a__minus₂(s₁(X), s₁(Y)) -> a__minus₂(mark₁(X), mark₁(Y))
a__quot₂(0, s₁(Y)) -> 0
a__quot₂(s₁(X), s₁(Y)) -> s₁(a__quot₂(a__minus₂(mark₁(X), mark₁(Y)), s₁(mark₁(Y))))
a__zWquot₂(XS, nil) -> nil
a__zWquot₂(nil, XS) -> nil
a__zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(a__quot₂(mark₁(X), mark₁(Y)), zWquot₂(XS, YS))
mark₁(from₁(X)) -> a__from₁(mark₁(X))
mark₁(sel₂(X1, X2)) -> a__sel₂(mark₁(X1), mark₁(X2))
mark₁(minus₂(X1, X2)) -> a__minus₂(mark₁(X1), mark₁(X2))
mark₁(quot₂(X1, X2)) -> a__quot₂(mark₁(X1), mark₁(X2))
mark₁(zWquot₂(X1, X2)) -> a__zWquot₂(mark₁(X1), mark₁(X2))
mark₁(cons₂(X1, X2)) -> cons₂(mark₁(X1), X2)
mark₁(s₁(X)) -> s₁(mark₁(X))
mark₁(0) -> 0
mark₁(nil) -> nil
a__from₁(X) -> from₁(X)
a__sel₂(X1, X2) -> sel₂(X1, X2)
a__minus₂(X1, X2) -> minus₂(X1, X2)
a__quot₂(X1, X2) -> quot₂(X1, X2)
a__zWquot₂(X1, X2) -> zWquot₂(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:

MARK₁(zWquot₂(X1, X2)) -> MARK₁(X2)
A__SEL₂(0, cons₂(X, XS)) -> MARK₁(X)
A__ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> MARK₁(Y)
MARK₁(sel₂(X1, X2)) -> A__SEL₂(mark₁(X1), mark₁(X2))
A__MINUS₂(s₁(X), s₁(Y)) -> MARK₁(Y)
A__SEL₂(s₁(N), cons₂(X, XS)) -> A__SEL₂(mark₁(N), mark₁(XS))
MARK₁(sel₂(X1, X2)) -> MARK₁(X1)
A__QUOT₂(s₁(X), s₁(Y)) -> MARK₁(Y)
A__QUOT₂(s₁(X), s₁(Y)) -> A__QUOT₂(a__minus₂(mark₁(X), mark₁(Y)), s₁(mark₁(Y)))
MARK₁(quot₂(X1, X2)) -> MARK₁(X1)
MARK₁(from₁(X)) -> MARK₁(X)
MARK₁(from₁(X)) -> A__FROM₁(mark₁(X))
MARK₁(cons₂(X1, X2)) -> MARK₁(X1)
A__SEL₂(s₁(N), cons₂(X, XS)) -> MARK₁(N)
MARK₁(quot₂(X1, X2)) -> MARK₁(X2)
MARK₁(minus₂(X1, X2)) -> MARK₁(X2)
A__QUOT₂(s₁(X), s₁(Y)) -> A__MINUS₂(mark₁(X), mark₁(Y))
MARK₁(minus₂(X1, X2)) -> A__MINUS₂(mark₁(X1), mark₁(X2))
MARK₁(quot₂(X1, X2)) -> A__QUOT₂(mark₁(X1), mark₁(X2))
MARK₁(minus₂(X1, X2)) -> MARK₁(X1)
A__MINUS₂(s₁(X), s₁(Y)) -> A__MINUS₂(mark₁(X), mark₁(Y))
MARK₁(sel₂(X1, X2)) -> MARK₁(X2)
A__SEL₂(s₁(N), cons₂(X, XS)) -> MARK₁(XS)
A__ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> A__QUOT₂(mark₁(X), mark₁(Y))
MARK₁(s₁(X)) -> MARK₁(X)
MARK₁(zWquot₂(X1, X2)) -> A__ZWQUOT₂(mark₁(X1), mark₁(X2))
MARK₁(zWquot₂(X1, X2)) -> MARK₁(X1)
A__QUOT₂(s₁(X), s₁(Y)) -> MARK₁(X)
A__ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> MARK₁(X)
A__MINUS₂(s₁(X), s₁(Y)) -> MARK₁(X)
A__FROM₁(X) -> MARK₁(X)

The TRS R consists of the following rules:

a__from₁(X) -> cons₂(mark₁(X), from₁(s₁(X)))
a__sel₂(0, cons₂(X, XS)) -> mark₁(X)
a__sel₂(s₁(N), cons₂(X, XS)) -> a__sel₂(mark₁(N), mark₁(XS))
a__minus₂(X, 0) -> 0
a__minus₂(s₁(X), s₁(Y)) -> a__minus₂(mark₁(X), mark₁(Y))
a__quot₂(0, s₁(Y)) -> 0
a__quot₂(s₁(X), s₁(Y)) -> s₁(a__quot₂(a__minus₂(mark₁(X), mark₁(Y)), s₁(mark₁(Y))))
a__zWquot₂(XS, nil) -> nil
a__zWquot₂(nil, XS) -> nil
a__zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(a__quot₂(mark₁(X), mark₁(Y)), zWquot₂(XS, YS))
mark₁(from₁(X)) -> a__from₁(mark₁(X))
mark₁(sel₂(X1, X2)) -> a__sel₂(mark₁(X1), mark₁(X2))
mark₁(minus₂(X1, X2)) -> a__minus₂(mark₁(X1), mark₁(X2))
mark₁(quot₂(X1, X2)) -> a__quot₂(mark₁(X1), mark₁(X2))
mark₁(zWquot₂(X1, X2)) -> a__zWquot₂(mark₁(X1), mark₁(X2))
mark₁(cons₂(X1, X2)) -> cons₂(mark₁(X1), X2)
mark₁(s₁(X)) -> s₁(mark₁(X))
mark₁(0) -> 0
mark₁(nil) -> nil
a__from₁(X) -> from₁(X)
a__sel₂(X1, X2) -> sel₂(X1, X2)
a__minus₂(X1, X2) -> minus₂(X1, X2)
a__quot₂(X1, X2) -> quot₂(X1, X2)
a__zWquot₂(X1, X2) -> zWquot₂(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

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

MARK₁(zWquot₂(X1, X2)) -> MARK₁(X2)
A__SEL₂(0, cons₂(X, XS)) -> MARK₁(X)
A__ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> MARK₁(Y)
MARK₁(sel₂(X1, X2)) -> A__SEL₂(mark₁(X1), mark₁(X2))
A__MINUS₂(s₁(X), s₁(Y)) -> MARK₁(Y)
A__SEL₂(s₁(N), cons₂(X, XS)) -> A__SEL₂(mark₁(N), mark₁(XS))
MARK₁(sel₂(X1, X2)) -> MARK₁(X1)
A__QUOT₂(s₁(X), s₁(Y)) -> MARK₁(Y)
A__QUOT₂(s₁(X), s₁(Y)) -> A__QUOT₂(a__minus₂(mark₁(X), mark₁(Y)), s₁(mark₁(Y)))
MARK₁(quot₂(X1, X2)) -> MARK₁(X1)
MARK₁(from₁(X)) -> MARK₁(X)
MARK₁(from₁(X)) -> A__FROM₁(mark₁(X))
MARK₁(cons₂(X1, X2)) -> MARK₁(X1)
A__SEL₂(s₁(N), cons₂(X, XS)) -> MARK₁(N)
MARK₁(quot₂(X1, X2)) -> MARK₁(X2)
MARK₁(minus₂(X1, X2)) -> MARK₁(X2)
A__QUOT₂(s₁(X), s₁(Y)) -> A__MINUS₂(mark₁(X), mark₁(Y))
MARK₁(minus₂(X1, X2)) -> A__MINUS₂(mark₁(X1), mark₁(X2))
MARK₁(quot₂(X1, X2)) -> A__QUOT₂(mark₁(X1), mark₁(X2))
MARK₁(minus₂(X1, X2)) -> MARK₁(X1)
A__MINUS₂(s₁(X), s₁(Y)) -> A__MINUS₂(mark₁(X), mark₁(Y))
MARK₁(sel₂(X1, X2)) -> MARK₁(X2)
A__SEL₂(s₁(N), cons₂(X, XS)) -> MARK₁(XS)
A__ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> A__QUOT₂(mark₁(X), mark₁(Y))
MARK₁(s₁(X)) -> MARK₁(X)
MARK₁(zWquot₂(X1, X2)) -> A__ZWQUOT₂(mark₁(X1), mark₁(X2))
MARK₁(zWquot₂(X1, X2)) -> MARK₁(X1)
A__QUOT₂(s₁(X), s₁(Y)) -> MARK₁(X)
A__ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> MARK₁(X)
A__MINUS₂(s₁(X), s₁(Y)) -> MARK₁(X)
A__FROM₁(X) -> MARK₁(X)

The TRS R consists of the following rules:

a__from₁(X) -> cons₂(mark₁(X), from₁(s₁(X)))
a__sel₂(0, cons₂(X, XS)) -> mark₁(X)
a__sel₂(s₁(N), cons₂(X, XS)) -> a__sel₂(mark₁(N), mark₁(XS))
a__minus₂(X, 0) -> 0
a__minus₂(s₁(X), s₁(Y)) -> a__minus₂(mark₁(X), mark₁(Y))
a__quot₂(0, s₁(Y)) -> 0
a__quot₂(s₁(X), s₁(Y)) -> s₁(a__quot₂(a__minus₂(mark₁(X), mark₁(Y)), s₁(mark₁(Y))))
a__zWquot₂(XS, nil) -> nil
a__zWquot₂(nil, XS) -> nil
a__zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(a__quot₂(mark₁(X), mark₁(Y)), zWquot₂(XS, YS))
mark₁(from₁(X)) -> a__from₁(mark₁(X))
mark₁(sel₂(X1, X2)) -> a__sel₂(mark₁(X1), mark₁(X2))
mark₁(minus₂(X1, X2)) -> a__minus₂(mark₁(X1), mark₁(X2))
mark₁(quot₂(X1, X2)) -> a__quot₂(mark₁(X1), mark₁(X2))
mark₁(zWquot₂(X1, X2)) -> a__zWquot₂(mark₁(X1), mark₁(X2))
mark₁(cons₂(X1, X2)) -> cons₂(mark₁(X1), X2)
mark₁(s₁(X)) -> s₁(mark₁(X))
mark₁(0) -> 0
mark₁(nil) -> nil
a__from₁(X) -> from₁(X)
a__sel₂(X1, X2) -> sel₂(X1, X2)
a__minus₂(X1, X2) -> minus₂(X1, X2)
a__quot₂(X1, X2) -> quot₂(X1, X2)
a__zWquot₂(X1, X2) -> zWquot₂(X1, X2)

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