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:

ap₂(ap₂(g, x), y) -> y
ap₂(f, x) -> ap₂(f, app₂(g, x))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

ap₂(ap₂(g, x), y) -> y
ap₂(f, x) -> ap₂(f, app₂(g, x))

Q is empty.

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

AP₂(f, x) -> AP₂(f, app₂(g, x))

The TRS R consists of the following rules:

ap₂(ap₂(g, x), y) -> y
ap₂(f, x) -> ap₂(f, app₂(g, x))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

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

AP₂(f, x) -> AP₂(f, app₂(g, x))

The TRS R consists of the following rules:

ap₂(ap₂(g, x), y) -> y
ap₂(f, x) -> ap₂(f, app₂(g, x))

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