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(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive Path Order [RPO].
Precedence:

activate₁ > f₁ > [g₁, n_h1, n_f1, h₁]

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

f(X) → g(n__h(n__f(X)))
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

(2) Obligation:

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

h(X) → n__h(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive Path Order [RPO].
Precedence:

h₁ > n_h1

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

h(X) → n__h(X)

(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(6) TRUE