(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
pi1 > [2ndspos2, 2ndsneg2] > rcons2
pi1 > [2ndspos2, 2ndsneg2] > activate1 > from1 > [cons2, s1, ncons2, posrecip1, negrecip1] > ns1
pi1 > [2ndspos2, 2ndsneg2] > activate1 > from1 > nfrom1
pi1 > 0 > rnil
square1 > times2 > plus2 > [cons2, s1, ncons2, posrecip1, negrecip1] > ns1
Status:
from1: multiset
cons2: multiset
nfrom1: multiset
ns1: multiset
2ndspos2: [1,2]
0: multiset
rnil: multiset
s1: multiset
ncons2: multiset
rcons2: [2,1]
posrecip1: [1]
activate1: multiset
2ndsneg2: [1,2]
negrecip1: [1]
pi1: multiset
plus2: multiset
times2: multiset
square1: 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)
s(X) → n__s(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:
cons(X1, X2) → n__cons(X1, X2)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
cons2 > ncons2
Status:
cons2: multiset
ncons2: 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)
(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