Termination w.r.t. Q proof of TRS/TRCSREx6_Luc98

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:

first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DirectTerminationProof

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

first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(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:

[first₂, activate_1] > nil > [s₁, cons₂, n_{_{from_1]}}
[first₂, activate_1] > n_{_first₂} > [s₁, cons₂, n_{_{from_1]}}
[first₂, activate_1] > from₁ > [s₁, cons₂, n_{_{from_1]}}
0 > [s₁, cons₂, n_{_{from_1]}}

Status:

from₁: [1]
first₂: [2,1]
n_{_from₁}: [1]
0: multiset
s₁: [1]
nil: multiset
cons₂: [2,1]
n_{_first₂}: [2,1]
activate₁: [1]