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:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(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:

CONS2(mark1(X1), X2) -> CONS2(X1, X2)
TOP1(mark1(X)) -> PROPER1(X)
ACTIVE1(zeros) -> CONS2(0, zeros)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
TAIL1(mark1(X)) -> TAIL1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
TAIL1(ok1(X)) -> TAIL1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(tail1(X)) -> PROPER1(X)
TOP1(ok1(X)) -> ACTIVE1(X)
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
TOP1(ok1(X)) -> TOP1(active1(X))
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
TOP1(mark1(X)) -> TOP1(proper1(X))
PROPER1(tail1(X)) -> TAIL1(proper1(X))
ACTIVE1(tail1(X)) -> TAIL1(active1(X))
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(tail1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(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:

CONS2(mark1(X1), X2) -> CONS2(X1, X2)
TOP1(mark1(X)) -> PROPER1(X)
ACTIVE1(zeros) -> CONS2(0, zeros)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
TAIL1(mark1(X)) -> TAIL1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
TAIL1(ok1(X)) -> TAIL1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(tail1(X)) -> PROPER1(X)
TOP1(ok1(X)) -> ACTIVE1(X)
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
TOP1(ok1(X)) -> TOP1(active1(X))
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
TOP1(mark1(X)) -> TOP1(proper1(X))
PROPER1(tail1(X)) -> TAIL1(proper1(X))
ACTIVE1(tail1(X)) -> TAIL1(active1(X))
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(tail1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 5 SCCs with 7 less nodes.

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

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

TAIL1(mark1(X)) -> TAIL1(X)
TAIL1(ok1(X)) -> TAIL1(X)

The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(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.


TAIL1(mark1(X)) -> TAIL1(X)
The remaining pairs can at least be oriented weakly.

TAIL1(ok1(X)) -> TAIL1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( TAIL1(x1) ) = max{0, x1 - 1}


POL( mark1(x1) ) = x1 + 2


POL( ok1(x1) ) = x1



The following usable rules [14] were oriented: none



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

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

TAIL1(ok1(X)) -> TAIL1(X)

The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(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.


TAIL1(ok1(X)) -> TAIL1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( TAIL1(x1) ) = max{0, x1 - 1}


POL( ok1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

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

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

CONS2(mark1(X1), X2) -> CONS2(X1, X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)

The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(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.


CONS2(mark1(X1), X2) -> CONS2(X1, X2)
The remaining pairs can at least be oriented weakly.

CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( CONS2(x1, x2) ) = max{0, x1 - 1}


POL( mark1(x1) ) = x1 + 2


POL( ok1(x1) ) = x1



The following usable rules [14] were oriented: none



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

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

CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)

The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(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.


CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( CONS2(x1, x2) ) = max{0, x2 - 1}


POL( ok1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

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

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

PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(tail1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(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.


PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.

PROPER1(tail1(X)) -> PROPER1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


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


POL( tail1(x1) ) = x1



The following usable rules [14] were oriented: none



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

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

PROPER1(tail1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(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.


PROPER1(tail1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( tail1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

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

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

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(tail1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(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.


ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
The remaining pairs can at least be oriented weakly.

ACTIVE1(tail1(X)) -> ACTIVE1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE1(x1) ) = max{0, x1 - 1}


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


POL( tail1(x1) ) = x1



The following usable rules [14] were oriented: none



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

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

ACTIVE1(tail1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(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.


ACTIVE1(tail1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE1(x1) ) = max{0, x1 - 1}


POL( tail1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ PisEmptyProof
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

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

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

TOP1(ok1(X)) -> TOP1(active1(X))
TOP1(mark1(X)) -> TOP1(proper1(X))

The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(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.


TOP1(mark1(X)) -> TOP1(proper1(X))
The remaining pairs can at least be oriented weakly.

TOP1(ok1(X)) -> TOP1(active1(X))
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( TOP1(x1) ) = x1


POL( mark1(x1) ) = x1 + 1


POL( proper1(x1) ) = x1


POL( ok1(x1) ) = x1


POL( active1(x1) ) = x1


POL( tail1(x1) ) = x1 + 2


POL( cons2(x1, x2) ) = max{0, 2x1 + x2 - 1}


POL( zeros ) = 1


POL( 0 ) = 0



The following usable rules [14] were oriented:

cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
proper1(0) -> ok1(0)
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(tail1(X)) -> tail1(active1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(tail1(X)) -> tail1(proper1(X))
tail1(ok1(X)) -> ok1(tail1(X))
proper1(zeros) -> ok1(zeros)



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

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

TOP1(ok1(X)) -> TOP1(active1(X))

The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(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.


TOP1(ok1(X)) -> TOP1(active1(X))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( TOP1(x1) ) = max{0, x1 - 1}


POL( ok1(x1) ) = 2


POL( active1(x1) ) = 0


POL( mark1(x1) ) = 0


POL( cons2(x1, x2) ) = x1


POL( tail1(x1) ) = x1



The following usable rules [14] were oriented:

cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(tail1(X)) -> tail1(active1(X))
tail1(ok1(X)) -> ok1(tail1(X))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active1(zeros) -> mark1(cons2(0, zeros))
active1(tail1(cons2(X, XS))) -> mark1(XS)
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(tail1(X)) -> tail1(active1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
tail1(mark1(X)) -> mark1(tail1(X))
proper1(zeros) -> ok1(zeros)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(tail1(X)) -> tail1(proper1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
tail1(ok1(X)) -> ok1(tail1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.