Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

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.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:

pi₁ > [2ndspos₂, 2ndsneg₂] > rnil > [n_s1, s₁]
pi₁ > [2ndspos₂, 2ndsneg₂] > posrecip₁ > [n_s1, s₁]
pi₁ > [2ndspos₂, 2ndsneg₂] > activate₁ > from₁ > [cons₂, n_cons2] > rcons₂ > [n_s1, s₁]
pi₁ > [2ndspos₂, 2ndsneg₂] > activate₁ > from₁ > [cons₂, n_cons2] > negrecip₁ > [n_s1, s₁]
pi₁ > [2ndspos₂, 2ndsneg₂] > activate₁ > from₁ > n_from1 > [n_s1, s₁]
pi₁ > 0 > rnil > [n_s1, s₁]
square₁ > times₂ > plus₂ > [n_s1, s₁]

Status:

from₁: multiset
plus₂: multiset
rnil: multiset
negrecip₁: multiset
n_cons2: multiset
rcons₂: multiset
posrecip₁: multiset
activate₁: multiset
n_s1: multiset
pi₁: [1]
square₁: multiset
0: multiset
2ndsneg₂: [1,2]
cons₂: multiset
2ndspos₂: [1,2]
n_from1: [1]
times₂: [1,2]
s₁: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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)
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

(2) Obligation:

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

s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:

s₁ > n_s1
[cons₂, n_cons2]

Status:

cons₂: [2,1]
n_cons2: [2,1]
s₁: multiset
n_s1: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

s(X) → n__s(X)

(4) Obligation:

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

cons(X1, X2) → n__cons(X1, X2)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:

cons₂ > n_cons2

Status:

cons₂: multiset
n_cons2: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

cons(X1, X2) → n__cons(X1, X2)

(6) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(7) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(8) TRUE