Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar16/SK90SLASH2.57.trs

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

f₂(g₁(h₂(x, y)), f₂(a, a)) -> f₂(h₂(x, x), g₁(f₂(y, a)))

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

f₂(g₁(h₂(x, y)), f₂(a, a)) -> f₂(h₂(x, x), g₁(f₂(y, a)))

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:

f₂(g₁(h₂(x, y)), f₂(a, a)) -> f₂(h₂(x, x), g₁(f₂(y, a)))

The set Q consists of the following terms:

f₂(g₁(h₂(x0, x1)), f₂(a, a))

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

F₂(g₁(h₂(x, y)), f₂(a, a)) -> F₂(h₂(x, x), g₁(f₂(y, a)))
F₂(g₁(h₂(x, y)), f₂(a, a)) -> F₂(y, a)

The TRS R consists of the following rules:

f₂(g₁(h₂(x, y)), f₂(a, a)) -> f₂(h₂(x, x), g₁(f₂(y, a)))

The set Q consists of the following terms:

f₂(g₁(h₂(x0, x1)), f₂(a, a))

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

F₂(g₁(h₂(x, y)), f₂(a, a)) -> F₂(h₂(x, x), g₁(f₂(y, a)))
F₂(g₁(h₂(x, y)), f₂(a, a)) -> F₂(y, a)

The TRS R consists of the following rules:

f₂(g₁(h₂(x, y)), f₂(a, a)) -> f₂(h₂(x, x), g₁(f₂(y, a)))

The set Q consists of the following terms:

f₂(g₁(h₂(x0, x1)), f₂(a, a))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 2 less nodes.