Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

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

f(f(X)) → f(g(f(g(f(X)))))
f(g(f(X))) → f(g(X))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

F(f(X)) → F(g(f(g(f(X)))))
F(f(X)) → F(g(f(X)))
F(g(f(X))) → F(g(X))

The TRS R consists of the following rules:

f(f(X)) → f(g(f(g(f(X)))))
f(g(f(X))) → f(g(X))

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(4) Obligation:

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

F(g(f(X))) → F(g(X))

The TRS R consists of the following rules:

f(f(X)) → f(g(f(g(f(X)))))
f(g(f(X))) → f(g(X))

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

F(g(f(X))) → F(g(X))

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[g₁, f₁] > F₁

Status:

F₁: multiset
g₁: multiset
f₁: multiset

The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

f(f(X)) → f(g(f(g(f(X)))))
f(g(f(X))) → f(g(X))

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

(7) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(8) TRUE