Termination w.r.t. Q proof of TRS/TRCSREx3_12_Luc96a

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:

from(X) → cons(X, n__from(n__s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
s(X) → n__s(X)
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:

from(X) → cons(X, n__from(n__s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
s(X) → n__s(X)
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 [19].
Precedence:

sel₂ > activate₁ > from₁ > cons₂ > n_{_s₁}
sel₂ > activate₁ > from₁ > n_{_from₁} > n_{_s₁}
sel₂ > activate₁ > s₁ > n_{_s₁}
0 > n_{_s₁}