Termination w.r.t. Q proof of TRS/TRCSREx1_2_Luc02c

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:

2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(n__s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DirectTerminationProof

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

2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(n__s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(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:

[2nd₁, activate_1] > from₁ > cons₂ > [n_{_cons₂}, n_{_from₁}, n_{_{s_1]}}
[2nd₁, activate_1] > s₁ > [n_{_cons₂}, n_{_from₁}, n_{_{s_1]}}

Status:

from₁: [1]
n_{_from₁}: [1]
n_{_s₁}: [1]
s₁: [1]
cons₂: [1,2]
activate₁: [1]
n_{_cons₂}: [2,1]
2nd₁: [1]