Termination w.r.t. Q proof of TRS/TRCSREx1_2_AEL03

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)))
2ndspos(0, Z) → rnil
2ndspos(s(N), cons(X, n__cons(Y, Z))) → rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) → rnil
2ndsneg(s(N), cons(X, n__cons(Y, Z))) → rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z)))
pi(X) → 2ndspos(X, from(0))
plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
times(0, Y) → 0
times(s(X), Y) → plus(Y, times(X, Y))
square(X) → times(X, X)
from(X) → n__from(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
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)))
2ndspos(0, Z) → rnil
2ndspos(s(N), cons(X, n__cons(Y, Z))) → rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) → rnil
2ndsneg(s(N), cons(X, n__cons(Y, Z))) → rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z)))
pi(X) → 2ndspos(X, from(0))
plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
times(0, Y) → 0
times(s(X), Y) → plus(Y, times(X, Y))
square(X) → times(X, X)
from(X) → n__from(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(X) → X

Q is empty.

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

Lexicographic Path Order [19].
Precedence:

pi₁ > [2ndspos₂, 2ndsneg_2] > rnil
pi₁ > [2ndspos₂, 2ndsneg_2] > rcons₂
pi₁ > [2ndspos₂, 2ndsneg_2] > posrecip₁
pi₁ > [2ndspos₂, 2ndsneg_2] > activate₁ > from₁ > cons₂ > n_{_cons₂}
pi₁ > [2ndspos₂, 2ndsneg_2] > activate₁ > from₁ > n_{_from₁}
pi₁ > [2ndspos₂, 2ndsneg_2] > activate₁ > from₁ > n_{_s₁}
pi₁ > [2ndspos₂, 2ndsneg_2] > activate₁ > s₁ > n_{_s₁}
pi₁ > [2ndspos₂, 2ndsneg_2] > negrecip₁
pi₁ > 0
square₁ > times₂ > plus₂ > s₁ > n_{_s₁}