Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

a__app2(nil, YS) -> mark1(YS)
a__app2(cons2(X, XS), YS) -> cons2(mark1(X), app2(XS, YS))
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
a__zWadr2(nil, YS) -> nil
a__zWadr2(XS, nil) -> nil
a__zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(a__app2(mark1(Y), cons2(mark1(X), nil)), zWadr2(XS, YS))
a__prefix1(L) -> cons2(nil, zWadr2(L, prefix1(L)))
mark1(app2(X1, X2)) -> a__app2(mark1(X1), mark1(X2))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(zWadr2(X1, X2)) -> a__zWadr2(mark1(X1), mark1(X2))
mark1(prefix1(X)) -> a__prefix1(mark1(X))
mark1(nil) -> nil
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__app2(X1, X2) -> app2(X1, X2)
a__from1(X) -> from1(X)
a__zWadr2(X1, X2) -> zWadr2(X1, X2)
a__prefix1(X) -> prefix1(X)

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

a__app2(nil, YS) -> mark1(YS)
a__app2(cons2(X, XS), YS) -> cons2(mark1(X), app2(XS, YS))
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
a__zWadr2(nil, YS) -> nil
a__zWadr2(XS, nil) -> nil
a__zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(a__app2(mark1(Y), cons2(mark1(X), nil)), zWadr2(XS, YS))
a__prefix1(L) -> cons2(nil, zWadr2(L, prefix1(L)))
mark1(app2(X1, X2)) -> a__app2(mark1(X1), mark1(X2))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(zWadr2(X1, X2)) -> a__zWadr2(mark1(X1), mark1(X2))
mark1(prefix1(X)) -> a__prefix1(mark1(X))
mark1(nil) -> nil
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__app2(X1, X2) -> app2(X1, X2)
a__from1(X) -> from1(X)
a__zWadr2(X1, X2) -> zWadr2(X1, X2)
a__prefix1(X) -> prefix1(X)

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:

A__ZWADR2(cons2(X, XS), cons2(Y, YS)) -> MARK1(X)
MARK1(app2(X1, X2)) -> A__APP2(mark1(X1), mark1(X2))
MARK1(prefix1(X)) -> MARK1(X)
MARK1(zWadr2(X1, X2)) -> MARK1(X1)
A__APP2(nil, YS) -> MARK1(YS)
MARK1(zWadr2(X1, X2)) -> MARK1(X2)
A__ZWADR2(cons2(X, XS), cons2(Y, YS)) -> MARK1(Y)
A__ZWADR2(cons2(X, XS), cons2(Y, YS)) -> A__APP2(mark1(Y), cons2(mark1(X), nil))
MARK1(app2(X1, X2)) -> MARK1(X1)
MARK1(s1(X)) -> MARK1(X)
MARK1(prefix1(X)) -> A__PREFIX1(mark1(X))
MARK1(zWadr2(X1, X2)) -> A__ZWADR2(mark1(X1), mark1(X2))
MARK1(from1(X)) -> MARK1(X)
A__APP2(cons2(X, XS), YS) -> MARK1(X)
MARK1(from1(X)) -> A__FROM1(mark1(X))
MARK1(cons2(X1, X2)) -> MARK1(X1)
A__FROM1(X) -> MARK1(X)
MARK1(app2(X1, X2)) -> MARK1(X2)

The TRS R consists of the following rules:

a__app2(nil, YS) -> mark1(YS)
a__app2(cons2(X, XS), YS) -> cons2(mark1(X), app2(XS, YS))
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
a__zWadr2(nil, YS) -> nil
a__zWadr2(XS, nil) -> nil
a__zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(a__app2(mark1(Y), cons2(mark1(X), nil)), zWadr2(XS, YS))
a__prefix1(L) -> cons2(nil, zWadr2(L, prefix1(L)))
mark1(app2(X1, X2)) -> a__app2(mark1(X1), mark1(X2))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(zWadr2(X1, X2)) -> a__zWadr2(mark1(X1), mark1(X2))
mark1(prefix1(X)) -> a__prefix1(mark1(X))
mark1(nil) -> nil
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__app2(X1, X2) -> app2(X1, X2)
a__from1(X) -> from1(X)
a__zWadr2(X1, X2) -> zWadr2(X1, X2)
a__prefix1(X) -> prefix1(X)

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

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ DependencyGraphProof

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

A__ZWADR2(cons2(X, XS), cons2(Y, YS)) -> MARK1(X)
MARK1(app2(X1, X2)) -> A__APP2(mark1(X1), mark1(X2))
MARK1(prefix1(X)) -> MARK1(X)
MARK1(zWadr2(X1, X2)) -> MARK1(X1)
A__APP2(nil, YS) -> MARK1(YS)
MARK1(zWadr2(X1, X2)) -> MARK1(X2)
A__ZWADR2(cons2(X, XS), cons2(Y, YS)) -> MARK1(Y)
A__ZWADR2(cons2(X, XS), cons2(Y, YS)) -> A__APP2(mark1(Y), cons2(mark1(X), nil))
MARK1(app2(X1, X2)) -> MARK1(X1)
MARK1(s1(X)) -> MARK1(X)
MARK1(prefix1(X)) -> A__PREFIX1(mark1(X))
MARK1(zWadr2(X1, X2)) -> A__ZWADR2(mark1(X1), mark1(X2))
MARK1(from1(X)) -> MARK1(X)
A__APP2(cons2(X, XS), YS) -> MARK1(X)
MARK1(from1(X)) -> A__FROM1(mark1(X))
MARK1(cons2(X1, X2)) -> MARK1(X1)
A__FROM1(X) -> MARK1(X)
MARK1(app2(X1, X2)) -> MARK1(X2)

The TRS R consists of the following rules:

a__app2(nil, YS) -> mark1(YS)
a__app2(cons2(X, XS), YS) -> cons2(mark1(X), app2(XS, YS))
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
a__zWadr2(nil, YS) -> nil
a__zWadr2(XS, nil) -> nil
a__zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(a__app2(mark1(Y), cons2(mark1(X), nil)), zWadr2(XS, YS))
a__prefix1(L) -> cons2(nil, zWadr2(L, prefix1(L)))
mark1(app2(X1, X2)) -> a__app2(mark1(X1), mark1(X2))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(zWadr2(X1, X2)) -> a__zWadr2(mark1(X1), mark1(X2))
mark1(prefix1(X)) -> a__prefix1(mark1(X))
mark1(nil) -> nil
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__app2(X1, X2) -> app2(X1, X2)
a__from1(X) -> from1(X)
a__zWadr2(X1, X2) -> zWadr2(X1, X2)
a__prefix1(X) -> prefix1(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
QDP
          ↳ QDPOrderProof

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

A__ZWADR2(cons2(X, XS), cons2(Y, YS)) -> MARK1(X)
MARK1(app2(X1, X2)) -> A__APP2(mark1(X1), mark1(X2))
MARK1(zWadr2(X1, X2)) -> MARK1(X1)
MARK1(prefix1(X)) -> MARK1(X)
A__APP2(nil, YS) -> MARK1(YS)
MARK1(zWadr2(X1, X2)) -> MARK1(X2)
A__ZWADR2(cons2(X, XS), cons2(Y, YS)) -> MARK1(Y)
A__ZWADR2(cons2(X, XS), cons2(Y, YS)) -> A__APP2(mark1(Y), cons2(mark1(X), nil))
MARK1(app2(X1, X2)) -> MARK1(X1)
MARK1(s1(X)) -> MARK1(X)
MARK1(zWadr2(X1, X2)) -> A__ZWADR2(mark1(X1), mark1(X2))
MARK1(from1(X)) -> MARK1(X)
MARK1(from1(X)) -> A__FROM1(mark1(X))
A__APP2(cons2(X, XS), YS) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)
A__FROM1(X) -> MARK1(X)
MARK1(app2(X1, X2)) -> MARK1(X2)

The TRS R consists of the following rules:

a__app2(nil, YS) -> mark1(YS)
a__app2(cons2(X, XS), YS) -> cons2(mark1(X), app2(XS, YS))
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
a__zWadr2(nil, YS) -> nil
a__zWadr2(XS, nil) -> nil
a__zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(a__app2(mark1(Y), cons2(mark1(X), nil)), zWadr2(XS, YS))
a__prefix1(L) -> cons2(nil, zWadr2(L, prefix1(L)))
mark1(app2(X1, X2)) -> a__app2(mark1(X1), mark1(X2))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(zWadr2(X1, X2)) -> a__zWadr2(mark1(X1), mark1(X2))
mark1(prefix1(X)) -> a__prefix1(mark1(X))
mark1(nil) -> nil
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__app2(X1, X2) -> app2(X1, X2)
a__from1(X) -> from1(X)
a__zWadr2(X1, X2) -> zWadr2(X1, X2)
a__prefix1(X) -> prefix1(X)

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__ZWADR2(cons2(X, XS), cons2(Y, YS)) -> MARK1(X)
MARK1(zWadr2(X1, X2)) -> MARK1(X1)
MARK1(prefix1(X)) -> MARK1(X)
MARK1(zWadr2(X1, X2)) -> MARK1(X2)
A__ZWADR2(cons2(X, XS), cons2(Y, YS)) -> MARK1(Y)
A__ZWADR2(cons2(X, XS), cons2(Y, YS)) -> A__APP2(mark1(Y), cons2(mark1(X), nil))
MARK1(s1(X)) -> MARK1(X)
MARK1(zWadr2(X1, X2)) -> A__ZWADR2(mark1(X1), mark1(X2))
MARK1(from1(X)) -> MARK1(X)
MARK1(from1(X)) -> A__FROM1(mark1(X))
A__APP2(cons2(X, XS), YS) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)
The remaining pairs can at least be oriented weakly.

MARK1(app2(X1, X2)) -> A__APP2(mark1(X1), mark1(X2))
A__APP2(nil, YS) -> MARK1(YS)
MARK1(app2(X1, X2)) -> MARK1(X1)
A__FROM1(X) -> MARK1(X)
MARK1(app2(X1, X2)) -> MARK1(X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( from1(x1) ) = 2x1 + 2


POL( a__app2(x1, x2) ) = 2x1 + x2 + 1


POL( mark1(x1) ) = x1


POL( a__zWadr2(x1, x2) ) = x1 + 2x2 + 2


POL( a__prefix1(x1) ) = 2x1 + 2


POL( A__APP2(x1, x2) ) = 2x1 + 2x2


POL( MARK1(x1) ) = max{0, 2x1 - 2}


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


POL( zWadr2(x1, x2) ) = x1 + 2x2 + 2


POL( cons2(x1, x2) ) = x1 + 2


POL( app2(x1, x2) ) = 2x1 + x2 + 1


POL( a__from1(x1) ) = 2x1 + 2


POL( s1(x1) ) = 2x1 + 2


POL( A__ZWADR2(x1, x2) ) = max{0, 2x1 + 2x2 - 2}


POL( A__FROM1(x1) ) = 2x1


POL( prefix1(x1) ) = 2x1 + 2



The following usable rules [14] were oriented:

a__prefix1(X) -> prefix1(X)
mark1(s1(X)) -> s1(mark1(X))
a__prefix1(L) -> cons2(nil, zWadr2(L, prefix1(L)))
a__zWadr2(X1, X2) -> zWadr2(X1, X2)
a__app2(cons2(X, XS), YS) -> cons2(mark1(X), app2(XS, YS))
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
mark1(app2(X1, X2)) -> a__app2(mark1(X1), mark1(X2))
a__app2(nil, YS) -> mark1(YS)
mark1(nil) -> nil
a__from1(X) -> from1(X)
a__zWadr2(nil, YS) -> nil
mark1(from1(X)) -> a__from1(mark1(X))
mark1(prefix1(X)) -> a__prefix1(mark1(X))
a__zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(a__app2(mark1(Y), cons2(mark1(X), nil)), zWadr2(XS, YS))
a__app2(X1, X2) -> app2(X1, X2)
a__zWadr2(XS, nil) -> nil
mark1(zWadr2(X1, X2)) -> a__zWadr2(mark1(X1), mark1(X2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
QDP
              ↳ DependencyGraphProof

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

MARK1(app2(X1, X2)) -> MARK1(X1)
MARK1(app2(X1, X2)) -> A__APP2(mark1(X1), mark1(X2))
A__APP2(nil, YS) -> MARK1(YS)
A__FROM1(X) -> MARK1(X)
MARK1(app2(X1, X2)) -> MARK1(X2)

The TRS R consists of the following rules:

a__app2(nil, YS) -> mark1(YS)
a__app2(cons2(X, XS), YS) -> cons2(mark1(X), app2(XS, YS))
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
a__zWadr2(nil, YS) -> nil
a__zWadr2(XS, nil) -> nil
a__zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(a__app2(mark1(Y), cons2(mark1(X), nil)), zWadr2(XS, YS))
a__prefix1(L) -> cons2(nil, zWadr2(L, prefix1(L)))
mark1(app2(X1, X2)) -> a__app2(mark1(X1), mark1(X2))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(zWadr2(X1, X2)) -> a__zWadr2(mark1(X1), mark1(X2))
mark1(prefix1(X)) -> a__prefix1(mark1(X))
mark1(nil) -> nil
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__app2(X1, X2) -> app2(X1, X2)
a__from1(X) -> from1(X)
a__zWadr2(X1, X2) -> zWadr2(X1, X2)
a__prefix1(X) -> prefix1(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
QDP
                  ↳ QDPOrderProof

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

MARK1(app2(X1, X2)) -> MARK1(X1)
MARK1(app2(X1, X2)) -> A__APP2(mark1(X1), mark1(X2))
A__APP2(nil, YS) -> MARK1(YS)
MARK1(app2(X1, X2)) -> MARK1(X2)

The TRS R consists of the following rules:

a__app2(nil, YS) -> mark1(YS)
a__app2(cons2(X, XS), YS) -> cons2(mark1(X), app2(XS, YS))
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
a__zWadr2(nil, YS) -> nil
a__zWadr2(XS, nil) -> nil
a__zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(a__app2(mark1(Y), cons2(mark1(X), nil)), zWadr2(XS, YS))
a__prefix1(L) -> cons2(nil, zWadr2(L, prefix1(L)))
mark1(app2(X1, X2)) -> a__app2(mark1(X1), mark1(X2))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(zWadr2(X1, X2)) -> a__zWadr2(mark1(X1), mark1(X2))
mark1(prefix1(X)) -> a__prefix1(mark1(X))
mark1(nil) -> nil
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__app2(X1, X2) -> app2(X1, X2)
a__from1(X) -> from1(X)
a__zWadr2(X1, X2) -> zWadr2(X1, X2)
a__prefix1(X) -> prefix1(X)

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.


MARK1(app2(X1, X2)) -> MARK1(X1)
MARK1(app2(X1, X2)) -> A__APP2(mark1(X1), mark1(X2))
MARK1(app2(X1, X2)) -> MARK1(X2)
The remaining pairs can at least be oriented weakly.

A__APP2(nil, YS) -> MARK1(YS)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( a__app2(x1, x2) ) = 2x1 + 2x2 + 1


POL( from1(x1) ) = x1 + 1


POL( mark1(x1) ) = x1


POL( a__zWadr2(x1, x2) ) = 2x1 + 2


POL( a__prefix1(x1) ) = max{0, -1}


POL( A__APP2(x1, x2) ) = 2x2 + 2


POL( MARK1(x1) ) = 2x1 + 2


POL( nil ) = 2


POL( zWadr2(x1, x2) ) = 2x1 + 2


POL( cons2(x1, x2) ) = max{0, -2}


POL( app2(x1, x2) ) = 2x1 + 2x2 + 1


POL( a__from1(x1) ) = x1 + 1


POL( s1(x1) ) = max{0, -2}


POL( prefix1(x1) ) = 0



The following usable rules [14] were oriented:

a__prefix1(X) -> prefix1(X)
mark1(s1(X)) -> s1(mark1(X))
a__prefix1(L) -> cons2(nil, zWadr2(L, prefix1(L)))
a__zWadr2(X1, X2) -> zWadr2(X1, X2)
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
a__app2(cons2(X, XS), YS) -> cons2(mark1(X), app2(XS, YS))
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
mark1(app2(X1, X2)) -> a__app2(mark1(X1), mark1(X2))
a__app2(nil, YS) -> mark1(YS)
mark1(nil) -> nil
a__from1(X) -> from1(X)
a__zWadr2(nil, YS) -> nil
mark1(from1(X)) -> a__from1(mark1(X))
mark1(prefix1(X)) -> a__prefix1(mark1(X))
a__zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(a__app2(mark1(Y), cons2(mark1(X), nil)), zWadr2(XS, YS))
a__app2(X1, X2) -> app2(X1, X2)
a__zWadr2(XS, nil) -> nil
mark1(zWadr2(X1, X2)) -> a__zWadr2(mark1(X1), mark1(X2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
QDP
                      ↳ DependencyGraphProof

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

A__APP2(nil, YS) -> MARK1(YS)

The TRS R consists of the following rules:

a__app2(nil, YS) -> mark1(YS)
a__app2(cons2(X, XS), YS) -> cons2(mark1(X), app2(XS, YS))
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
a__zWadr2(nil, YS) -> nil
a__zWadr2(XS, nil) -> nil
a__zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(a__app2(mark1(Y), cons2(mark1(X), nil)), zWadr2(XS, YS))
a__prefix1(L) -> cons2(nil, zWadr2(L, prefix1(L)))
mark1(app2(X1, X2)) -> a__app2(mark1(X1), mark1(X2))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(zWadr2(X1, X2)) -> a__zWadr2(mark1(X1), mark1(X2))
mark1(prefix1(X)) -> a__prefix1(mark1(X))
mark1(nil) -> nil
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__app2(X1, X2) -> app2(X1, X2)
a__from1(X) -> from1(X)
a__zWadr2(X1, X2) -> zWadr2(X1, X2)
a__prefix1(X) -> prefix1(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 1 less node.