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(empty, l) -> l
f2(cons2(x, k), l) -> g3(k, l, cons2(x, k))
g3(a, b, c) -> f2(a, cons2(b, c))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

f2(empty, l) -> l
f2(cons2(x, k), l) -> g3(k, l, cons2(x, k))
g3(a, b, c) -> f2(a, cons2(b, c))

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(x, k), l) -> G3(k, l, cons2(x, k))
G3(a, b, c) -> F2(a, cons2(b, c))

The TRS R consists of the following rules:

f2(empty, l) -> l
f2(cons2(x, k), l) -> g3(k, l, cons2(x, k))
g3(a, b, c) -> f2(a, cons2(b, c))

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(x, k), l) -> G3(k, l, cons2(x, k))
G3(a, b, c) -> F2(a, cons2(b, c))

The TRS R consists of the following rules:

f2(empty, l) -> l
f2(cons2(x, k), l) -> g3(k, l, cons2(x, k))
g3(a, b, c) -> f2(a, cons2(b, c))

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(x, k), l) -> G3(k, l, cons2(x, k))
The remaining pairs can at least be oriented weakly.

G3(a, b, c) -> F2(a, cons2(b, c))
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( G3(x1, ..., x3) ) = 2x1 + x3


POL( cons2(x1, x2) ) = 3x1 + 2x2 + 3


POL( F2(x1, x2) ) = max{0, 2x1 - 2}



The following usable rules [14] were oriented: none



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

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

G3(a, b, c) -> F2(a, cons2(b, c))

The TRS R consists of the following rules:

f2(empty, l) -> l
f2(cons2(x, k), l) -> g3(k, l, cons2(x, k))
g3(a, b, c) -> f2(a, cons2(b, c))

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.