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(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.
↳ QTRS
↳ DirectTerminationProof
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.
We use [23] with the following order to prove termination.
Recursive path order with status [2].
Quasi-Precedence:
pi1 > [2ndspos2, 2ndsneg2] > rnil > [nfrom1, s1, ncons2, rcons2]
pi1 > [2ndspos2, 2ndsneg2] > posrecip1 > [nfrom1, s1, ncons2, rcons2]
pi1 > [2ndspos2, 2ndsneg2] > activate1 > from1 > [cons2, negrecip1] > [nfrom1, s1, ncons2, rcons2]
pi1 > 0 > [nfrom1, s1, ncons2, rcons2]
square1 > times2 > plus2 > [nfrom1, s1, ncons2, rcons2]
Status: from1: multiset
nfrom1: multiset
rcons2: multiset
0: multiset
negrecip1: multiset
times2: multiset
cons2: [2,1]
square1: multiset
2ndspos2: [1,2]
activate1: [1]
posrecip1: multiset
plus2: multiset
2ndsneg2: [1,2]
s1: multiset
ncons2: [2,1]
pi1: multiset
rnil: multiset