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:

rev12(0, nil) -> 0
rev12(s1(X), nil) -> s1(X)
rev12(X, cons2(Y, L)) -> rev12(Y, L)
rev1(nil) -> nil
rev1(cons2(X, L)) -> cons2(rev12(X, L), rev22(X, L))
rev22(X, nil) -> nil
rev22(X, cons2(Y, L)) -> rev1(cons2(X, rev1(rev22(Y, L))))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

rev12(0, nil) -> 0
rev12(s1(X), nil) -> s1(X)
rev12(X, cons2(Y, L)) -> rev12(Y, L)
rev1(nil) -> nil
rev1(cons2(X, L)) -> cons2(rev12(X, L), rev22(X, L))
rev22(X, nil) -> nil
rev22(X, cons2(Y, L)) -> rev1(cons2(X, rev1(rev22(Y, L))))

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:

REV22(X, cons2(Y, L)) -> REV1(cons2(X, rev1(rev22(Y, L))))
REV22(X, cons2(Y, L)) -> REV22(Y, L)
REV1(cons2(X, L)) -> REV22(X, L)
REV1(cons2(X, L)) -> REV12(X, L)
REV22(X, cons2(Y, L)) -> REV1(rev22(Y, L))
REV12(X, cons2(Y, L)) -> REV12(Y, L)

The TRS R consists of the following rules:

rev12(0, nil) -> 0
rev12(s1(X), nil) -> s1(X)
rev12(X, cons2(Y, L)) -> rev12(Y, L)
rev1(nil) -> nil
rev1(cons2(X, L)) -> cons2(rev12(X, L), rev22(X, L))
rev22(X, nil) -> nil
rev22(X, cons2(Y, L)) -> rev1(cons2(X, rev1(rev22(Y, L))))

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:

REV22(X, cons2(Y, L)) -> REV1(cons2(X, rev1(rev22(Y, L))))
REV22(X, cons2(Y, L)) -> REV22(Y, L)
REV1(cons2(X, L)) -> REV22(X, L)
REV1(cons2(X, L)) -> REV12(X, L)
REV22(X, cons2(Y, L)) -> REV1(rev22(Y, L))
REV12(X, cons2(Y, L)) -> REV12(Y, L)

The TRS R consists of the following rules:

rev12(0, nil) -> 0
rev12(s1(X), nil) -> s1(X)
rev12(X, cons2(Y, L)) -> rev12(Y, L)
rev1(nil) -> nil
rev1(cons2(X, L)) -> cons2(rev12(X, L), rev22(X, L))
rev22(X, nil) -> nil
rev22(X, cons2(Y, L)) -> rev1(cons2(X, rev1(rev22(Y, L))))

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, cons2(Y, L)) -> REV12(Y, L)

The TRS R consists of the following rules:

rev12(0, nil) -> 0
rev12(s1(X), nil) -> s1(X)
rev12(X, cons2(Y, L)) -> rev12(Y, L)
rev1(nil) -> nil
rev1(cons2(X, L)) -> cons2(rev12(X, L), rev22(X, L))
rev22(X, nil) -> nil
rev22(X, cons2(Y, L)) -> rev1(cons2(X, rev1(rev22(Y, L))))

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, cons2(Y, L)) -> REV12(Y, L)
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( cons2(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:

rev12(0, nil) -> 0
rev12(s1(X), nil) -> s1(X)
rev12(X, cons2(Y, L)) -> rev12(Y, L)
rev1(nil) -> nil
rev1(cons2(X, L)) -> cons2(rev12(X, L), rev22(X, L))
rev22(X, nil) -> nil
rev22(X, cons2(Y, L)) -> rev1(cons2(X, rev1(rev22(Y, L))))

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, cons2(Y, L)) -> REV1(cons2(X, rev1(rev22(Y, L))))
REV22(X, cons2(Y, L)) -> REV22(Y, L)
REV1(cons2(X, L)) -> REV22(X, L)
REV22(X, cons2(Y, L)) -> REV1(rev22(Y, L))

The TRS R consists of the following rules:

rev12(0, nil) -> 0
rev12(s1(X), nil) -> s1(X)
rev12(X, cons2(Y, L)) -> rev12(Y, L)
rev1(nil) -> nil
rev1(cons2(X, L)) -> cons2(rev12(X, L), rev22(X, L))
rev22(X, nil) -> nil
rev22(X, cons2(Y, L)) -> rev1(cons2(X, rev1(rev22(Y, L))))

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, cons2(Y, L)) -> REV22(Y, L)
REV1(cons2(X, L)) -> REV22(X, L)
REV22(X, cons2(Y, L)) -> REV1(rev22(Y, L))
The remaining pairs can at least be oriented weakly.

REV22(X, cons2(Y, L)) -> REV1(cons2(X, rev1(rev22(Y, L))))
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( rev22(x1, x2) ) = x1 + x2


POL( REV22(x1, x2) ) = max{0, 2x1 + 2x2 - 2}


POL( rev1(x1) ) = x1


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


POL( REV1(x1) ) = max{0, 2x1 - 2}


POL( 0 ) = max{0, -3}


POL( s1(x1) ) = 0


POL( nil ) = max{0, -3}


POL( cons2(x1, x2) ) = x1 + x2 + 3



The following usable rules [14] were oriented:

rev1(cons2(X, L)) -> cons2(rev12(X, L), rev22(X, L))
rev12(0, nil) -> 0
rev12(X, cons2(Y, L)) -> rev12(Y, L)
rev22(X, nil) -> nil
rev12(s1(X), nil) -> s1(X)
rev1(nil) -> nil
rev22(X, cons2(Y, L)) -> rev1(cons2(X, rev1(rev22(Y, L))))



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

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

REV22(X, cons2(Y, L)) -> REV1(cons2(X, rev1(rev22(Y, L))))

The TRS R consists of the following rules:

rev12(0, nil) -> 0
rev12(s1(X), nil) -> s1(X)
rev12(X, cons2(Y, L)) -> rev12(Y, L)
rev1(nil) -> nil
rev1(cons2(X, L)) -> cons2(rev12(X, L), rev22(X, L))
rev22(X, nil) -> nil
rev22(X, cons2(Y, L)) -> rev1(cons2(X, rev1(rev22(Y, L))))

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.