(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
from(X) → cons(X, n__from(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)
cons(X1, X2) → n__cons(X1, X2)
activate(n__from(X)) → from(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(X) → X
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Combined order from the following AFS and order.
from(
x1) =
from(
x1)
cons(
x1,
x2) =
cons(
x1,
x2)
n__from(
x1) =
n__from(
x1)
s(
x1) =
s(
x1)
2ndspos(
x1,
x2) =
2ndspos(
x1,
x2)
0 =
0
rnil =
rnil
n__cons(
x1,
x2) =
n__cons(
x1,
x2)
rcons(
x1,
x2) =
rcons(
x1,
x2)
posrecip(
x1) =
x1
activate(
x1) =
activate(
x1)
2ndsneg(
x1,
x2) =
2ndsneg(
x1,
x2)
negrecip(
x1) =
x1
pi(
x1) =
pi(
x1)
plus(
x1,
x2) =
plus(
x1,
x2)
times(
x1,
x2) =
times(
x1,
x2)
square(
x1) =
square(
x1)
Recursive path order with status [RPO].
Quasi-Precedence:
pi1 > 0 > [2ndspos2, rnil, 2ndsneg2] > [from1, activate1] > cons2 > ncons2 > s1
pi1 > 0 > [2ndspos2, rnil, 2ndsneg2] > [from1, activate1] > nfrom1 > s1
pi1 > 0 > [2ndspos2, rnil, 2ndsneg2] > rcons2 > s1
square1 > times2 > 0 > [2ndspos2, rnil, 2ndsneg2] > [from1, activate1] > cons2 > ncons2 > s1
square1 > times2 > 0 > [2ndspos2, rnil, 2ndsneg2] > [from1, activate1] > nfrom1 > s1
square1 > times2 > 0 > [2ndspos2, rnil, 2ndsneg2] > rcons2 > s1
square1 > times2 > plus2 > s1
Status:
from1: multiset
cons2: multiset
nfrom1: multiset
s1: [1]
2ndspos2: [1,2]
0: multiset
rnil: multiset
ncons2: multiset
rcons2: multiset
activate1: multiset
2ndsneg2: [1,2]
pi1: multiset
plus2: multiset
times2: [1,2]
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(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)
cons(X1, X2) → n__cons(X1, X2)
activate(n__from(X)) → from(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(X) → X
(2) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(3) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(4) YES