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

Q is empty.

(1) DirectTerminationProof (EQUIVALENT transformation)

We use [DIRECT_TERMINATION] with the following order to prove termination.

Knuth-Bendix order [KBO] with precedence:

activate₁ > f₁ > c₁ > a > n_f1 > g₁ > n_g1 > n_a

and weight map:

a=4
n__a=1
f_1=2
c_1=1
n__f_1=2
n__g_1=2
g_1=2
activate_1=3

The variable weight is 1

(2) TRUE