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:

rev1(nil) -> nil
rev1(++2(x, y)) -> ++2(rev12(x, y), rev22(x, y))
rev12(x, nil) -> x
rev12(x, ++2(y, z)) -> rev12(y, z)
rev22(x, nil) -> nil
rev22(x, ++2(y, z)) -> rev1(++2(x, rev1(rev22(y, z))))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

rev1(nil) -> nil
rev1(++2(x, y)) -> ++2(rev12(x, y), rev22(x, y))
rev12(x, nil) -> x
rev12(x, ++2(y, z)) -> rev12(y, z)
rev22(x, nil) -> nil
rev22(x, ++2(y, z)) -> rev1(++2(x, rev1(rev22(y, z))))

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:

REV12(x, ++2(y, z)) -> REV12(y, z)
REV1(++2(x, y)) -> REV12(x, y)
REV22(x, ++2(y, z)) -> REV22(y, z)
REV22(x, ++2(y, z)) -> REV1(rev22(y, z))
REV22(x, ++2(y, z)) -> REV1(++2(x, rev1(rev22(y, z))))
REV1(++2(x, y)) -> REV22(x, y)

The TRS R consists of the following rules:

rev1(nil) -> nil
rev1(++2(x, y)) -> ++2(rev12(x, y), rev22(x, y))
rev12(x, nil) -> x
rev12(x, ++2(y, z)) -> rev12(y, z)
rev22(x, nil) -> nil
rev22(x, ++2(y, z)) -> rev1(++2(x, rev1(rev22(y, z))))

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:

REV12(x, ++2(y, z)) -> REV12(y, z)
REV1(++2(x, y)) -> REV12(x, y)
REV22(x, ++2(y, z)) -> REV22(y, z)
REV22(x, ++2(y, z)) -> REV1(rev22(y, z))
REV22(x, ++2(y, z)) -> REV1(++2(x, rev1(rev22(y, z))))
REV1(++2(x, y)) -> REV22(x, y)

The TRS R consists of the following rules:

rev1(nil) -> nil
rev1(++2(x, y)) -> ++2(rev12(x, y), rev22(x, y))
rev12(x, nil) -> x
rev12(x, ++2(y, z)) -> rev12(y, z)
rev22(x, nil) -> nil
rev22(x, ++2(y, z)) -> rev1(++2(x, rev1(rev22(y, z))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 1 less node.

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

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

REV12(x, ++2(y, z)) -> REV12(y, z)

The TRS R consists of the following rules:

rev1(nil) -> nil
rev1(++2(x, y)) -> ++2(rev12(x, y), rev22(x, y))
rev12(x, nil) -> x
rev12(x, ++2(y, z)) -> rev12(y, z)
rev22(x, nil) -> nil
rev22(x, ++2(y, z)) -> rev1(++2(x, rev1(rev22(y, z))))

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.


REV12(x, ++2(y, z)) -> REV12(y, z)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( REV12(x1, x2) ) = 3x1 + 2x2 + 3


POL( ++2(x1, x2) ) = 2x1 + 3x2 + 1



The following usable rules [14] were oriented: none



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

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

rev1(nil) -> nil
rev1(++2(x, y)) -> ++2(rev12(x, y), rev22(x, y))
rev12(x, nil) -> x
rev12(x, ++2(y, z)) -> rev12(y, z)
rev22(x, nil) -> nil
rev22(x, ++2(y, z)) -> rev1(++2(x, rev1(rev22(y, z))))

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

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

REV22(x, ++2(y, z)) -> REV22(y, z)
REV22(x, ++2(y, z)) -> REV1(rev22(y, z))
REV22(x, ++2(y, z)) -> REV1(++2(x, rev1(rev22(y, z))))
REV1(++2(x, y)) -> REV22(x, y)

The TRS R consists of the following rules:

rev1(nil) -> nil
rev1(++2(x, y)) -> ++2(rev12(x, y), rev22(x, y))
rev12(x, nil) -> x
rev12(x, ++2(y, z)) -> rev12(y, z)
rev22(x, nil) -> nil
rev22(x, ++2(y, z)) -> rev1(++2(x, rev1(rev22(y, z))))

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.


REV22(x, ++2(y, z)) -> REV22(y, z)
REV22(x, ++2(y, z)) -> REV1(rev22(y, z))
REV1(++2(x, y)) -> REV22(x, y)
The remaining pairs can at least be oriented weakly.

REV22(x, ++2(y, z)) -> REV1(++2(x, rev1(rev22(y, z))))
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( REV22(x1, x2) ) = 2x2 + 3


POL( rev22(x1, x2) ) = x2


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


POL( rev1(x1) ) = x1


POL( rev12(x1, x2) ) = max{0, 2x1 - 3}


POL( REV1(x1) ) = 2x1 + 3


POL( nil ) = 2



The following usable rules [14] were oriented:

rev22(x, ++2(y, z)) -> rev1(++2(x, rev1(rev22(y, z))))
rev1(nil) -> nil
rev22(x, nil) -> nil
rev1(++2(x, y)) -> ++2(rev12(x, y), rev22(x, y))



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

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

REV22(x, ++2(y, z)) -> REV1(++2(x, rev1(rev22(y, z))))

The TRS R consists of the following rules:

rev1(nil) -> nil
rev1(++2(x, y)) -> ++2(rev12(x, y), rev22(x, y))
rev12(x, nil) -> x
rev12(x, ++2(y, z)) -> rev12(y, z)
rev22(x, nil) -> nil
rev22(x, ++2(y, z)) -> rev1(++2(x, rev1(rev22(y, z))))

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.