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(app2(rec, t), u), v), 0) -> t
app2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> app2(app2(u, x), app2(app2(app2(app2(rec, t), u), v), x))
app2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> app2(app2(v, f), app2(app2(app2(app2(rectuv, t), u), v), app2(f, n)))
app2(app2(app2(app2(rectuv, t), u), v), n) -> app2(app2(app2(app2(rec, t), u), v), n)

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

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

app2(app2(app2(app2(rec, t), u), v), 0) -> t
app2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> app2(app2(u, x), app2(app2(app2(app2(rec, t), u), v), x))
app2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> app2(app2(v, f), app2(app2(app2(app2(rectuv, t), u), v), app2(f, n)))
app2(app2(app2(app2(rectuv, t), u), v), n) -> app2(app2(app2(app2(rec, t), u), v), n)

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(app2(rec, t), u), v), 0) -> t
app2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> app2(app2(u, x), app2(app2(app2(app2(rec, t), u), v), x))
app2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> app2(app2(v, f), app2(app2(app2(app2(rectuv, t), u), v), app2(f, n)))
app2(app2(app2(app2(rectuv, t), u), v), n) -> app2(app2(app2(app2(rec, t), u), v), n)

The set Q consists of the following terms:

app2(app2(app2(app2(rec, x0), x1), x2), 0)
app2(app2(app2(app2(rec, x0), x1), x2), app2(s, x3))
app2(app2(app2(app2(rec, x0), x1), x2), app2(lim, x3))
app2(app2(app2(app2(rectuv, x0), x1), x2), n)


Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

APP2(app2(app2(app2(rectuv, t), u), v), n) -> APP2(app2(rec, t), u)
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(app2(v, f), app2(app2(app2(app2(rectuv, t), u), v), app2(f, n)))
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(app2(rectuv, t), u)
APP2(app2(app2(app2(rectuv, t), u), v), n) -> APP2(app2(app2(app2(rec, t), u), v), n)
APP2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> APP2(app2(u, x), app2(app2(app2(app2(rec, t), u), v), x))
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(app2(app2(app2(rectuv, t), u), v), app2(f, n))
APP2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> APP2(u, x)
APP2(app2(app2(app2(rectuv, t), u), v), n) -> APP2(rec, t)
APP2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> APP2(app2(app2(app2(rec, t), u), v), x)
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(f, n)
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(app2(app2(rectuv, t), u), v)
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(v, f)
APP2(app2(app2(app2(rectuv, t), u), v), n) -> APP2(app2(app2(rec, t), u), v)
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(rectuv, t)

The TRS R consists of the following rules:

app2(app2(app2(app2(rec, t), u), v), 0) -> t
app2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> app2(app2(u, x), app2(app2(app2(app2(rec, t), u), v), x))
app2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> app2(app2(v, f), app2(app2(app2(app2(rectuv, t), u), v), app2(f, n)))
app2(app2(app2(app2(rectuv, t), u), v), n) -> app2(app2(app2(app2(rec, t), u), v), n)

The set Q consists of the following terms:

app2(app2(app2(app2(rec, x0), x1), x2), 0)
app2(app2(app2(app2(rec, x0), x1), x2), app2(s, x3))
app2(app2(app2(app2(rec, x0), x1), x2), app2(lim, x3))
app2(app2(app2(app2(rectuv, x0), x1), x2), n)

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:

APP2(app2(app2(app2(rectuv, t), u), v), n) -> APP2(app2(rec, t), u)
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(app2(v, f), app2(app2(app2(app2(rectuv, t), u), v), app2(f, n)))
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(app2(rectuv, t), u)
APP2(app2(app2(app2(rectuv, t), u), v), n) -> APP2(app2(app2(app2(rec, t), u), v), n)
APP2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> APP2(app2(u, x), app2(app2(app2(app2(rec, t), u), v), x))
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(app2(app2(app2(rectuv, t), u), v), app2(f, n))
APP2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> APP2(u, x)
APP2(app2(app2(app2(rectuv, t), u), v), n) -> APP2(rec, t)
APP2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> APP2(app2(app2(app2(rec, t), u), v), x)
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(f, n)
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(app2(app2(rectuv, t), u), v)
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(v, f)
APP2(app2(app2(app2(rectuv, t), u), v), n) -> APP2(app2(app2(rec, t), u), v)
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(rectuv, t)

The TRS R consists of the following rules:

app2(app2(app2(app2(rec, t), u), v), 0) -> t
app2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> app2(app2(u, x), app2(app2(app2(app2(rec, t), u), v), x))
app2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> app2(app2(v, f), app2(app2(app2(app2(rectuv, t), u), v), app2(f, n)))
app2(app2(app2(app2(rectuv, t), u), v), n) -> app2(app2(app2(app2(rec, t), u), v), n)

The set Q consists of the following terms:

app2(app2(app2(app2(rec, x0), x1), x2), 0)
app2(app2(app2(app2(rec, x0), x1), x2), app2(s, x3))
app2(app2(app2(app2(rec, x0), x1), x2), app2(lim, x3))
app2(app2(app2(app2(rectuv, x0), x1), x2), n)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 9 less nodes.

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

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

APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(app2(v, f), app2(app2(app2(app2(rectuv, t), u), v), app2(f, n)))
APP2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> APP2(u, x)
APP2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> APP2(app2(app2(app2(rec, t), u), v), x)
APP2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> APP2(app2(u, x), app2(app2(app2(app2(rec, t), u), v), x))
APP2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> APP2(v, f)

The TRS R consists of the following rules:

app2(app2(app2(app2(rec, t), u), v), 0) -> t
app2(app2(app2(app2(rec, t), u), v), app2(s, x)) -> app2(app2(u, x), app2(app2(app2(app2(rec, t), u), v), x))
app2(app2(app2(app2(rec, t), u), v), app2(lim, f)) -> app2(app2(v, f), app2(app2(app2(app2(rectuv, t), u), v), app2(f, n)))
app2(app2(app2(app2(rectuv, t), u), v), n) -> app2(app2(app2(app2(rec, t), u), v), n)

The set Q consists of the following terms:

app2(app2(app2(app2(rec, x0), x1), x2), 0)
app2(app2(app2(app2(rec, x0), x1), x2), app2(s, x3))
app2(app2(app2(app2(rec, x0), x1), x2), app2(lim, x3))
app2(app2(app2(app2(rectuv, x0), x1), x2), n)

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