Termination w.r.t. Q proof of TRS/higher-order/BirdEx2_8

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(const, x), y) → x
app(app(app(subst, f), g), x) → app(app(f, x), app(g, x))
app(app(fix, f), x) → app(app(f, app(fix, f)), x)

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

app(app(const, x), y) → x
app(app(app(subst, f), g), x) → app(app(f, x), app(g, x))
app(app(fix, f), x) → app(app(f, app(fix, f)), x)

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(const, x), y) → x
app(app(app(subst, f), g), x) → app(app(f, x), app(g, x))
app(app(fix, f), x) → app(app(f, app(fix, f)), x)

The set Q consists of the following terms:

app(app(const, x0), x1)
app(app(app(subst, x0), x1), x2)
app(app(fix, x0), x1)

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(fix, f), x) → APP(f, app(fix, f))
APP(app(fix, f), x) → APP(app(f, app(fix, f)), x)
APP(app(app(subst, f), g), x) → APP(f, x)
APP(app(app(subst, f), g), x) → APP(app(f, x), app(g, x))
APP(app(app(subst, f), g), x) → APP(g, x)

The TRS R consists of the following rules:

app(app(const, x), y) → x
app(app(app(subst, f), g), x) → app(app(f, x), app(g, x))
app(app(fix, f), x) → app(app(f, app(fix, f)), x)

The set Q consists of the following terms:

app(app(const, x0), x1)
app(app(app(subst, x0), x1), x2)
app(app(fix, x0), x1)

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

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

APP(app(fix, f), x) → APP(f, app(fix, f))
APP(app(fix, f), x) → APP(app(f, app(fix, f)), x)
APP(app(app(subst, f), g), x) → APP(f, x)
APP(app(app(subst, f), g), x) → APP(app(f, x), app(g, x))
APP(app(app(subst, f), g), x) → APP(g, x)

The TRS R consists of the following rules:

app(app(const, x), y) → x
app(app(app(subst, f), g), x) → app(app(f, x), app(g, x))
app(app(fix, f), x) → app(app(f, app(fix, f)), x)

The set Q consists of the following terms:

app(app(const, x0), x1)
app(app(app(subst, x0), x1), x2)
app(app(fix, x0), x1)

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