(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(n__s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
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:
[2nd1, activate1] > from1 > [cons2, ncons2]
[2nd1, activate1] > from1 > nfrom1
[2nd1, activate1] > from1 > [ns1, s1]
Status:
from1: multiset
cons2: multiset
nfrom1: multiset
2nd1: [1]
ncons2: multiset
activate1: [1]
s1: multiset
ns1: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(n__s(X)))
from(X) → n__from(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(cons(x1, x2)) = 1 + x1 + x2
POL(n__cons(x1, x2)) = x1 + x2
POL(n__s(x1)) = x1
POL(s(x1)) = 1 + x1
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)
s(X) → n__s(X)
(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