(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
[first2, nil, activate1] > from1 > [s1, cons2] > nfirst2
[first2, nil, activate1] > from1 > nfrom1 > nfirst2
0 > nfirst2
Status:
nfirst2: multiset
cons2: multiset
from1: multiset
nfrom1: multiset
s1: multiset
activate1: [1]
first2: [2,1]
0: multiset
nil: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(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) TRUE