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:

ap2(ap2(ff, x), x) -> ap2(ap2(x, ap2(ff, x)), ap2(ap2(cons, x), nil))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

ap2(ap2(ff, x), x) -> ap2(ap2(x, ap2(ff, x)), ap2(ap2(cons, x), nil))

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:

AP2(ap2(ff, x), x) -> AP2(ap2(cons, x), nil)
AP2(ap2(ff, x), x) -> AP2(ap2(x, ap2(ff, x)), ap2(ap2(cons, x), nil))
AP2(ap2(ff, x), x) -> AP2(x, ap2(ff, x))
AP2(ap2(ff, x), x) -> AP2(cons, x)

The TRS R consists of the following rules:

ap2(ap2(ff, x), x) -> ap2(ap2(x, ap2(ff, x)), ap2(ap2(cons, x), nil))

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:

AP2(ap2(ff, x), x) -> AP2(ap2(cons, x), nil)
AP2(ap2(ff, x), x) -> AP2(ap2(x, ap2(ff, x)), ap2(ap2(cons, x), nil))
AP2(ap2(ff, x), x) -> AP2(x, ap2(ff, x))
AP2(ap2(ff, x), x) -> AP2(cons, x)

The TRS R consists of the following rules:

ap2(ap2(ff, x), x) -> ap2(ap2(x, ap2(ff, x)), ap2(ap2(cons, x), nil))

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

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

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

AP2(ap2(ff, x), x) -> AP2(ap2(x, ap2(ff, x)), ap2(ap2(cons, x), nil))
AP2(ap2(ff, x), x) -> AP2(x, ap2(ff, x))

The TRS R consists of the following rules:

ap2(ap2(ff, x), x) -> ap2(ap2(x, ap2(ff, x)), ap2(ap2(cons, x), nil))

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.


AP2(ap2(ff, x), x) -> AP2(ap2(x, ap2(ff, x)), ap2(ap2(cons, x), nil))
AP2(ap2(ff, x), x) -> AP2(x, ap2(ff, x))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(AP2(x1, x2)) = x1 + 2·x1·x2   
POL(ap2(x1, x2)) = x1 + 2·x1·x2   
POL(cons) = 0   
POL(ff) = 2   
POL(nil) = 2   

The following usable rules [14] were oriented:

ap2(ap2(ff, x), x) -> ap2(ap2(x, ap2(ff, x)), ap2(ap2(cons, x), nil))



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

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

ap2(ap2(ff, x), x) -> ap2(ap2(x, ap2(ff, x)), ap2(ap2(cons, x), nil))

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.