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

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:

[a, activate₁] > [f₁, g₁, n_f1] > n_a

Status:

f₁: multiset
a: multiset
g₁: multiset
n_f1: multiset
activate₁: multiset
n_a: multiset

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

(2) Obligation:

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

f(X) → n__f(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:

f₁ > n_f1

Status:

f₁: multiset
n_f1: multiset

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

f(X) → n__f(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