Termination w.r.t. Q proof of TRS/TRCSRExConc_Zan97

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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.

↳ QTRS

  ↳ DirectTerminationProof

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.

We use [23] with the following order to prove termination.

Lexicographic path order with status [19].
Quasi-Precedence:

activate₁ > f₁ > g₁
activate₁ > f₁ > n_{_h₁}
activate₁ > f₁ > n_{_f₁}
activate₁ > h₁ > n_{_h₁}

Status:

h₁: [1]
f₁: [1]
n_{_h₁}: [1]
n_{_f₁}: [1]
g₁: [1]
activate₁: [1]