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:

f2(x, empty) -> x
f2(empty, cons2(a, k)) -> f2(cons2(a, k), k)
f2(cons2(a, k), y) -> f2(y, k)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

f2(x, empty) -> x
f2(empty, cons2(a, k)) -> f2(cons2(a, k), k)
f2(cons2(a, k), y) -> f2(y, k)

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:

F2(cons2(a, k), y) -> F2(y, k)
F2(empty, cons2(a, k)) -> F2(cons2(a, k), k)

The TRS R consists of the following rules:

f2(x, empty) -> x
f2(empty, cons2(a, k)) -> f2(cons2(a, k), k)
f2(cons2(a, k), y) -> f2(y, k)

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:

F2(cons2(a, k), y) -> F2(y, k)
F2(empty, cons2(a, k)) -> F2(cons2(a, k), k)

The TRS R consists of the following rules:

f2(x, empty) -> x
f2(empty, cons2(a, k)) -> f2(cons2(a, k), k)
f2(cons2(a, k), y) -> f2(y, k)

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.


F2(empty, cons2(a, k)) -> F2(cons2(a, k), k)
The remaining pairs can at least be oriented weakly.

F2(cons2(a, k), y) -> F2(y, k)
Used ordering: Polynomial interpretation [21]:

POL(F2(x1, x2)) = x1 + 2·x2   
POL(cons2(x1, x2)) = x1·x2 + 2·x2   
POL(empty) = 2   

The following usable rules [14] were oriented: none



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

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

F2(cons2(a, k), y) -> F2(y, k)

The TRS R consists of the following rules:

f2(x, empty) -> x
f2(empty, cons2(a, k)) -> f2(cons2(a, k), k)
f2(cons2(a, k), y) -> f2(y, k)

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.


F2(cons2(a, k), y) -> F2(y, k)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(F2(x1, x2)) = 2·x1 + 3·x1·x2 + 3·x2   
POL(cons2(x1, x2)) = 1 + 2·x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ PisEmptyProof

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

f2(x, empty) -> x
f2(empty, cons2(a, k)) -> f2(cons2(a, k), k)
f2(cons2(a, k), y) -> f2(y, k)

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.