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

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:

ap(ap(g, x), y) → y
ap(f, x) → ap(f, app(g, x))

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

ap(ap(g, x), y) → y
ap(f, x) → ap(f, app(g, 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:

ap(ap(g, x), y) → y
ap(f, x) → ap(f, app(g, x))

The set Q consists of the following terms:

ap(ap(g, x0), x1)
ap(f, x0)

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

AP(f, x) → AP(f, app(g, x))

The TRS R consists of the following rules:

ap(ap(g, x), y) → y
ap(f, x) → ap(f, app(g, x))

The set Q consists of the following terms:

ap(ap(g, x0), x1)
ap(f, x0)

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:

AP(f, x) → AP(f, app(g, x))

The TRS R consists of the following rules:

ap(ap(g, x), y) → y
ap(f, x) → ap(f, app(g, x))

The set Q consists of the following terms:

ap(ap(g, x0), x1)
ap(f, x0)

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