Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

app2(app2(app2(rec, f), x), 0) -> x
app2(app2(app2(rec, f), x), app2(s, y)) -> app2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

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

app2(app2(app2(rec, f), x), 0) -> x
app2(app2(app2(rec, f), x), app2(s, y)) -> app2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))

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:

app2(app2(app2(rec, f), x), 0) -> x
app2(app2(app2(rec, f), x), app2(s, y)) -> app2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))

The set Q consists of the following terms:

app2(app2(app2(rec, x0), x1), 0)
app2(app2(app2(rec, x0), x1), app2(s, x2))


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

APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(f, app2(s, y))
APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))
APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(app2(app2(rec, f), x), y)

The TRS R consists of the following rules:

app2(app2(app2(rec, f), x), 0) -> x
app2(app2(app2(rec, f), x), app2(s, y)) -> app2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))

The set Q consists of the following terms:

app2(app2(app2(rec, x0), x1), 0)
app2(app2(app2(rec, x0), x1), app2(s, x2))

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

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP

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

APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(f, app2(s, y))
APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))
APP2(app2(app2(rec, f), x), app2(s, y)) -> APP2(app2(app2(rec, f), x), y)

The TRS R consists of the following rules:

app2(app2(app2(rec, f), x), 0) -> x
app2(app2(app2(rec, f), x), app2(s, y)) -> app2(app2(f, app2(s, y)), app2(app2(app2(rec, f), x), y))

The set Q consists of the following terms:

app2(app2(app2(rec, x0), x1), 0)
app2(app2(app2(rec, x0), x1), app2(s, x2))

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