Termination w.r.t. Q proof of TRS/TRCSREx4_7_56_Bor03

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)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
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)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
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.

Recursive path order with status [2].
Quasi-Precedence:

after₂ > activate₁ > from₁ > cons₂ > n_{_from₁}
after₂ > activate₁ > from₁ > n_{_s₁} > n_{_from₁}
after₂ > activate₁ > s₁ > n_{_s₁} > n_{_from₁}
0 > n_{_from₁}

Status:

from₁: [1]
n_{_from₁}: multiset
after₂: [1,2]
n_{_s₁}: [1]
s₁: multiset
0: multiset
cons₂: multiset
activate₁: multiset