Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar16/RubioSLASHmfp90b.trs

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:

a -> g₁(c)
g₁(a) -> b
f₂(g₁(X), b) -> f₂(a, X)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a -> g₁(c)
g₁(a) -> b
f₂(g₁(X), b) -> f₂(a, 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:

F₂(g₁(X), b) -> A
F₂(g₁(X), b) -> F₂(a, X)
A -> G₁(c)

The TRS R consists of the following rules:

a -> g₁(c)
g₁(a) -> b
f₂(g₁(X), b) -> f₂(a, X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

F₂(g₁(X), b) -> A
F₂(g₁(X), b) -> F₂(a, X)
A -> G₁(c)

The TRS R consists of the following rules:

a -> g₁(c)
g₁(a) -> b
f₂(g₁(X), b) -> f₂(a, X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

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

F₂(g₁(X), b) -> F₂(a, X)

The TRS R consists of the following rules:

a -> g₁(c)
g₁(a) -> b
f₂(g₁(X), b) -> f₂(a, X)

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