Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar1/higher-orderSLASHAProVE

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

  ↳ Non-Overlap Check

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.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ 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))

The set Q consists of the following terms:

ap₂(ap₂(ff, x0), x0)

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))

The set Q consists of the following terms:

ap₂(ap₂(ff, x0), x0)

We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ 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))

The set Q consists of the following terms:

ap₂(ap₂(ff, x0), x0)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 0 SCCs with 4 less nodes.