(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] > rcons2 > [ns1, 0, rnil, s1]
pi1 > [2ndspos2, 2ndsneg2] > posrecip1 > [ns1, 0, rnil, s1]
pi1 > [2ndspos2, 2ndsneg2] > activate1 > from1 > [cons2, ncons2] > [ns1, 0, rnil, s1]
pi1 > [2ndspos2, 2ndsneg2] > activate1 > from1 > nfrom1 > [ns1, 0, rnil, s1]
pi1 > [2ndspos2, 2ndsneg2] > negrecip1 > [ns1, 0, rnil, s1]
square1 > times2 > plus2 > [ns1, 0, rnil, s1]

Status:
from1: multiset
cons2: multiset
nfrom1: multiset
ns1: multiset
2ndspos2: [1,2]
0: multiset
rnil: multiset
s1: multiset
ncons2: multiset
rcons2: [2,1]
posrecip1: multiset
activate1: multiset
2ndsneg2: [1,2]
negrecip1: multiset
pi1: multiset
plus2: [2,1]
times2: [2,1]
square1: [1]

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 > ns1
cons2 > ncons2

Status:
s1: [1]
ns1: [1]
cons2: [1,2]
ncons2: [1,2]

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)
cons(X1, X2) → n__cons(X1, X2)


(4) Obligation:

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

(5) RisEmptyProof (EQUIVALENT transformation)

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

(6) TRUE