Termination w.r.t. Q proof of TRS/higher-order/LifantsevEx7OrdinalRec.trs

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

  ↳ Overlay + Local Confluence

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.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ 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)

The set Q consists of the following terms:

app(app(app(app(rec, x0), x1), x2), 0)
app(app(app(app(rec, x0), x1), x2), app(s, x3))
app(app(app(app(rec, x0), x1), x2), app(lim, x3))
app(app(app(app(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:

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

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)

The set Q consists of the following terms:

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

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

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

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

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)

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

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

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

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)

The set Q consists of the following terms:

app(app(app(app(rec, x0), x1), x2), 0)
app(app(app(app(rec, x0), x1), x2), app(s, x3))
app(app(app(app(rec, x0), x1), x2), app(lim, x3))
app(app(app(app(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

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ 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(u, x)
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(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(v, f)

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)

The set Q consists of the following terms:

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

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