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:

app(app(app(app(rec, t), u), v), 0) → t
app(app(app(app(rec, t), u), v), app(s, x)) → app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) → app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) → app(app(app(app(rec, t), u), v), n)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

app(app(app(app(rec, t), u), v), 0) → t
app(app(app(app(rec, t), u), v), app(s, x)) → app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) → app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) → app(app(app(app(rec, t), u), v), n)

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

APP(app(app(app(rectuv, t), u), v), n) → APP(app(app(rec, t), u), v)
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(app(app(rectuv, t), u), v)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(v, f)
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rectuv, t), u), v), n) → APP(app(rec, t), u)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rectuv, t), u), v), n) → APP(rec, t)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(f, n)
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(u, x)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(rectuv, t)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(app(rectuv, t), u)
APP(app(app(app(rectuv, t), u), v), n) → APP(app(app(app(rec, t), u), v), n)

The TRS R consists of the following rules:

app(app(app(app(rec, t), u), v), 0) → t
app(app(app(app(rec, t), u), v), app(s, x)) → app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) → app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) → app(app(app(app(rec, t), u), v), n)

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:

APP(app(app(app(rectuv, t), u), v), n) → APP(app(app(rec, t), u), v)
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(app(app(rectuv, t), u), v)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(v, f)
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rectuv, t), u), v), n) → APP(app(rec, t), u)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rectuv, t), u), v), n) → APP(rec, t)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(f, n)
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(u, x)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(rectuv, t)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(app(rectuv, t), u)
APP(app(app(app(rectuv, t), u), v), n) → APP(app(app(app(rec, t), u), v), n)

The TRS R consists of the following rules:

app(app(app(app(rec, t), u), v), 0) → t
app(app(app(app(rec, t), u), v), app(s, x)) → app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) → app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) → app(app(app(app(rec, t), u), v), n)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 9 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP

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

APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(v, f)
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(app(u, x), app(app(app(app(rec, t), u), v), x))

The TRS R consists of the following rules:

app(app(app(app(rec, t), u), v), 0) → t
app(app(app(app(rec, t), u), v), app(s, x)) → app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) → app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) → app(app(app(app(rec, t), u), v), n)

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