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(f, a(f, x)) → a(x, g)
a(x, g) → a(f, a(g, a(f, x)))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

a(f, a(f, x)) → a(x, g)
a(x, g) → a(f, a(g, a(f, 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:

A(f, a(f, x)) → A(x, g)
A(x, g) → A(g, a(f, x))
A(x, g) → A(f, x)
A(x, g) → A(f, a(g, a(f, x)))

The TRS R consists of the following rules:

a(f, a(f, x)) → a(x, g)
a(x, g) → a(f, a(g, a(f, x)))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof

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

A(f, a(f, x)) → A(x, g)
A(x, g) → A(g, a(f, x))
A(x, g) → A(f, x)
A(x, g) → A(f, a(g, a(f, x)))

The TRS R consists of the following rules:

a(f, a(f, x)) → a(x, g)
a(x, g) → a(f, a(g, a(f, x)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof

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

A(f, a(f, x)) → A(x, g)
A(x, g) → A(g, a(f, x))
A(x, g) → A(f, x)
A(x, g) → A(f, a(g, a(f, x)))

The TRS R consists of the following rules:

a(f, a(f, x)) → a(x, g)
a(x, g) → a(f, a(g, a(f, 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 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP

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

A(f, a(f, x)) → A(x, g)
A(x, g) → A(f, x)
A(x, g) → A(f, a(g, a(f, x)))

The TRS R consists of the following rules:

a(f, a(f, x)) → a(x, g)
a(x, g) → a(f, a(g, a(f, x)))

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