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(a)) → f(g(n__f(a)))
f(X) → n__f(X)
activate(n__f(X)) → f(X)
activate(X) → X

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic Path Order [LPO].
Precedence:

activate₁ > f₁ > a
activate₁ > f₁ > g₁
activate₁ > f₁ > n_f1

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

f(f(a)) → f(g(n__f(a)))
f(X) → n__f(X)
activate(n__f(X)) → f(X)
activate(X) → X

(2) Obligation:

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

(3) RisEmptyProof (EQUIVALENT transformation)

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

(4) TRUE