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(const, x), y) -> x
app2(app2(app2(subst, f), g), x) -> app2(app2(f, x), app2(g, x))
app2(app2(fix, f), x) -> app2(app2(f, app2(fix, f)), x)

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

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

app2(app2(const, x), y) -> x
app2(app2(app2(subst, f), g), x) -> app2(app2(f, x), app2(g, x))
app2(app2(fix, f), x) -> app2(app2(f, app2(fix, f)), x)

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(const, x), y) -> x
app2(app2(app2(subst, f), g), x) -> app2(app2(f, x), app2(g, x))
app2(app2(fix, f), x) -> app2(app2(f, app2(fix, f)), x)

The set Q consists of the following terms:

app2(app2(const, x0), x1)
app2(app2(app2(subst, x0), x1), x2)
app2(app2(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:

APP2(app2(fix, f), x) -> APP2(f, app2(fix, f))
APP2(app2(fix, f), x) -> APP2(app2(f, app2(fix, f)), x)
APP2(app2(app2(subst, f), g), x) -> APP2(f, x)
APP2(app2(app2(subst, f), g), x) -> APP2(app2(f, x), app2(g, x))
APP2(app2(app2(subst, f), g), x) -> APP2(g, x)

The TRS R consists of the following rules:

app2(app2(const, x), y) -> x
app2(app2(app2(subst, f), g), x) -> app2(app2(f, x), app2(g, x))
app2(app2(fix, f), x) -> app2(app2(f, app2(fix, f)), x)

The set Q consists of the following terms:

app2(app2(const, x0), x1)
app2(app2(app2(subst, x0), x1), x2)
app2(app2(fix, x0), x1)

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(fix, f), x) -> APP2(f, app2(fix, f))
APP2(app2(fix, f), x) -> APP2(app2(f, app2(fix, f)), x)
APP2(app2(app2(subst, f), g), x) -> APP2(f, x)
APP2(app2(app2(subst, f), g), x) -> APP2(app2(f, x), app2(g, x))
APP2(app2(app2(subst, f), g), x) -> APP2(g, x)

The TRS R consists of the following rules:

app2(app2(const, x), y) -> x
app2(app2(app2(subst, f), g), x) -> app2(app2(f, x), app2(g, x))
app2(app2(fix, f), x) -> app2(app2(f, app2(fix, f)), x)

The set Q consists of the following terms:

app2(app2(const, x0), x1)
app2(app2(app2(subst, x0), x1), x2)
app2(app2(fix, x0), x1)

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