(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
from(X) → cons(X, n__from(n__s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
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(X) → X
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
sel2 > activate1 > from1 > [cons2, nfrom1, ns1]
sel2 > activate1 > s1 > [cons2, nfrom1, ns1]
0 > [cons2, nfrom1, ns1]
Status:
from1: [1]
cons2: [2,1]
nfrom1: [1]
ns1: multiset
sel2: [1,2]
0: multiset
s1: multiset
activate1: [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)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
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(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