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
      ↳ QDPOrderProof

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.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


A__QUOT2(s1(X), s1(Y)) -> A__QUOT2(a__minus2(mark1(X), mark1(Y)), s1(mark1(Y)))
The remaining pairs can at least be oriented weakly.

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)
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)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( from1(x1) ) = max{0, -1}


POL( sel2(x1, x2) ) = max{0, -1}


POL( minus2(x1, x2) ) = max{0, -1}


POL( mark1(x1) ) = 1


POL( a__zWquot2(x1, x2) ) = max{0, -1}


POL( A__ZWQUOT2(x1, x2) ) = 1


POL( A__SEL2(x1, x2) ) = 1


POL( quot2(x1, x2) ) = 1


POL( MARK1(x1) ) = 1


POL( zWquot2(x1, x2) ) = 0


POL( 0 ) = max{0, -1}


POL( a__minus2(x1, x2) ) = 0


POL( a__quot2(x1, x2) ) = 1


POL( A__MINUS2(x1, x2) ) = 1


POL( nil ) = 0


POL( cons2(x1, x2) ) = max{0, x1 - 1}


POL( A__QUOT2(x1, x2) ) = x1


POL( a__sel2(x1, x2) ) = 1


POL( a__from1(x1) ) = 1


POL( s1(x1) ) = 1


POL( A__FROM1(x1) ) = 1



The following usable rules [14] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
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)
MARK1(sel2(X1, X2)) -> A__SEL2(mark1(X1), mark1(X2))
A__ZWQUOT2(cons2(X, XS), cons2(Y, YS)) -> MARK1(Y)
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)
MARK1(quot2(X1, X2)) -> MARK1(X1)
MARK1(from1(X)) -> MARK1(X)
MARK1(from1(X)) -> A__FROM1(mark1(X))
A__SEL2(s1(N), cons2(X, XS)) -> MARK1(N)
MARK1(cons2(X1, X2)) -> MARK1(X1)
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__ZWQUOT2(cons2(X, XS), cons2(Y, YS)) -> MARK1(X)
A__QUOT2(s1(X), s1(Y)) -> 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.