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)) → c(n__f(g(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:
Recursive path order with status [RPO].
Quasi-Precedence:

[f₁, activate₁] > [a, c₁, g₁]
[f₁, activate₁] > n_f1

Status:

f₁: [1]
a: multiset
c₁: multiset
n_f1: multiset
g₁: multiset
activate₁: [1]

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

f(f(a)) → c(n__f(g(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